The new Architect? Yes or No?

Pros

• 1 credit cheaper so you can rez it for 3
• You can get back any card from Archives, Hedge Fund anyone?
• You can get 2 cards out of Archives
• It's not that good in Assett spam decks

Cons

• Trashable! This is a huge thing! Against Freedom Khumalo: Crypto-Anarchist I loved to have 2 ICE with the outermost being Architect so both ICE where protected against Hippo.
• You can't install 2 cards
• You can't peek into R&D
• The art is worse ;-) especially compared to the full bleed Architect but I like it!

I think this is a Sentry to go for HB, the face check is punishing from the early game, sometimes from Turn 1. It's cheap for a Sentry and usually gives you forward tempo, by installing a Rashida Jaheem again, a trashed ICE or defensive Upgrade like Ash 2X3ZB9CY.
I have mixed feelings about it compared to Architect. The fact that it costs 1 less is something, that it let's you get back a card to hand is huge, especially in these Fully Operational HB lists, but it's also nice to use it again for even a Hedge Fund as it pays back itself this way.
It's worse in asset decks, as it only gives you 1 install

​This card is bonkers! And makes any strategy that keeps servers open hard to play. And for a while now I haven't played against a deck without it.

Let's start with some actions...

1. Run 3x times, play Apocalypse wipe the board end of your turn install all your Crowdfundings from the bin and gain tempo compared to your opponent.
2. You can Run 3x, discard Crowdfunding at the end of your turn and install it from your heap (Nisei Rules 1.1, 12.2.2.2.e)
3. Play Hayley, Run 3x install Crowdfunding from the bin, trigger your ability and get a clickless install! This means you can't installed anything within your turn.
4. You can counter Scarcity of Resources when installing it from your heap, as it still costs 0!
5. You can sell Crowdfundings to Aesop's Pawnshop and reinstall them after Running 3x

It synergieses with everything that wants to run.
It works very well with Liza Talking Thunder: Prominent Legislator as you don't care when they trash your Crowdfundings. And of coure Gabriel Santiago: Consummate Professional. It plays well together with cards like: Paragon, Bankroll, Aumakua, Hot Pursuit, Diversion of Funds, Security Testing, Sneakdoor Beta. When you are a Shaper you can add: Takobi, Kyuban, Beth Kilrain-Chang as with 5 clicks 3 runs is much easier and Cyberdelia although it only pays you once per turn but make 3 runs much easier. As Anarch it works well with Bad Pub (Val), Yusuf, Datasucker and anything else that will load while running that is why you see a lot of them currently and anything that kill ICE like Hippo, Knifed, Spooned, Forked. And of course The Turning Wheel as well as Prey. Works well as a resource in tag me decks, as your opponent usually doesn't trash it.

It allows you to have an endless economy as long as you are able to run 3x within one turn, every like 3 turns. It's biggest enemy is Slot Machine or other cheap rez hard to break ICE like News Hound or Cortex Lock. And as a result Mti when you are not running Employee Strike. Not to forget Targeted Marketing and as a result Sol do you really want to give the corp 10 credits to get 3 & a card?

On the other hand, against a net damage deck like PU which plays 1000 cuts, you don't want to draw that much. In this cases you might don't like it drawing you cards and having your hand clogged with cards.

And of course, we get that sweet Alt Art in our Store Champ 2019 Kits!

​​This card reminds me a bit of SanSan City Grid, expensive to rez, expensive to trash and it enables fast advance and MCA Austerity Policy as it enables scoring 5/3 agendas within one turn and needs some loading time.

How does it work?
You can only "score" agendas that have the exact same agenda points as power counters on Lady Liberty. That means after rezzing at the end of the runners turn you can score a 3/1 the next turn, afterwards a 3/2 or 4/2 and then a 5/3, meaning you can use it to score 6 points over 3 turns. The advantage is, besides rezzing (5 creds) you don't pay anything but clicks and this way you would save the 12 credits needed to score 6 points (net 7 credits) and it has a high trash cost. From my point of view, when you can use it to score a 5/3 it was worth it's investment.
On the other side, you are not scoring the agenda, which means agendas with "When scored" abilities like Nisei MK II, Elective Upgrade or Project Atlas are not that valuable with this card, the agendas give you points but don't support the agenda ability. It works best with agendas without "When Scored" abilities, like: Global Food Initiative, Obokata Protocol, The Future Perfect or passive abilities like AR-Enhanced Security, Gila Hands Arcology and The Cleaners. Thus making NBN (Near-Earth Hub: Broadcast Center & NBN: Controlling the Message) a very strong contender for using the Lady. Additionally you should consider to run a mixed agenda composition to enable to use Lady Liberty as often as possible. On the other hand it gets useless when you have 4 power counter, but no agenda with 4 or more points within your deck.
What you need to consider as well, "Clot lock Shaper" is not an issue, as Clot is preventing you from scoring and you are not scoring agendas anymore (and there is no install), the same applies for The Source and Traffic Jam as it's increasing advancement requirements. To sum it up, strategies that prevent fast advance don't work against Lady Liberty! On the other hand, an Employee Strike (or any other current) will stay active. So in this regards it's like a Film Critic for corps, you will not be able to deactivate currents this way.
It makes the most sense in a deck that can rez it quite cheap e.g. Mumba Temple or put it in a heavy ICEed remote. So the Runner gets broke when trashing it and as a result opening a scoring windows. So in this regards it's comparable to MCA Austerity Policy. And in an asset heavy deck you might be able to recur it with Team Sponsorship, but that would mean the runner has thrashed it or you overwrite it and score an agenda the hard way to be able to recur it. For recursion you could also use Preemptive Action but how to find it again? Or you use Restore which is fine with a 1x card!
The question is how to use it? Do you have the agendas in hand? That means the runner can snipe them out of your hand or do you hope to draw them? Both has it's risk. Are you able to defend your remote, your HQ and R&D at the same time? When you want to draw the agendas some filter like Daily Business Show or additional draw like Rashida Jaheem, (and of course the banned Sensie Actors Union or Estelle Moon) come in handy. On top of that, having Jeeves Model Bioroids enables you to use the Lady and have another click for doing things, which puts pressure on the runner.
The downside is, it's 1x per Deck restriction, so you need to have cards to find it, like Tech Startup but this means telling your opponent what you plan and they can prepare for it and the Startups are cheap to trash. And you can only rez & load your Lady in the next turn. Or you use it as a must trash Asset which is 1x per Deck and if you find it fine, but it's an additional way to win. It's influence free. That is a plus.
Full Jank with Helium-3 Deposit is possible, so you can score Government Takeover or Vanity Project. e.g. You need to score Helium-3 Deposit the hard way, have the Lady in a server, so when you score it, you can place 2 power counter on it. When you rez it end of the runners turn, next turn you have 3 power counters on it and can score a 5/3 next turn. Or it's ticking already and you immediately are in Vanity or Government Takeover range, will you have it in you deck? Can you fetch it with Fast Track + Biotic Labor to win the game? Or you have your 4x Advanced Helium-3 Deposit and rez the Lady end of Runners turn, so you can score it, place the power counter on the Lady and then you use your 3 counters to place a 5/3 from hand in your score area.
The ID where it might shine the most is Blue Sun: Powering the Future assuming it's not Employee Striked. It can be used to score Agendas and then pull it back into hand to start from scratch with little payment (1 click) and of course the slot in the remote server.

District 99 enables you to get cards back from your heap, based on it's design (you need to load it) you want to see it early, because later in the game the value goes down very much. On the other hand you don't want to play 3x in your deck. So usually you use it as 1x which generates value when you can find it early, otherwise you don't use it. But how to load it?
The most obvious possibilities are MaxX: Maximum Punk Rock (yes her ability is loading District 99) or Inject (only trashing programs) and Patchwork that was also released in R&R. I think outside of MaxX the use is to niche to reliably load it, so I can't see other IDs using it. On top of Maxx it should be paired with Patchwork so you can trash cards to Patchwork to load your District which makes it more reliable.
On top of that, the strong dominance of the bin breakers (Paperclip, Black Orchestra & MKUltra) makes all of the mentioned cards strong! They allow you neat tricks, like run on a server, install a program from your heap, using the discount from Patchwork by trashing another bin breaker. Get the missing breaker in the bin in case you need it & get your District counter. Without the bin breakers, e.g. for rotation, I can't see a proper use case scenario. So as a summary, this cards best use case is Maxx & Patchwork.

Of course there is a chance that the corp is triggering District 99, e.g. with trash a program from Archer, Rototurret, Ichi 1.0 or others. But it's not reliable enough to justify to run this in your deck. Next Question...

What to recur with it? Only cards in faction, so the obvious choices are cards that MaxX: Maximum Punk Rock milled, that might be important for you, like Patchwork, Ice Carver or to replay Hacktivist Meeting. Or economy cards to reuse them again. Lot's of economy cards are neutral, that leaves Liberated Account, Stimhack, Day Job or Guinea Pig. Other options are: D4v1d, DDoS and The Archivist.
Splashing it into other factions doesn't seem to be that beneficial, but in theory you could pair it with Aesop's Pawnshop but this would need enough programs and hardware to sell to it. Or maybe a criminal deck with those terminal directive breakers (Abagnale, Lustig and Demara) and or Spy Camera and Sports Hopper hopper to have further Diversion of Funds and recursion. But that already seems janky.

I've tried to make this card work (in a competitive environment), without success in a Patchwork Maxx deck, but I'm not that experienced in playing Maxx too. Personally it never felt good to use 2 clicks to get Liberated, install it again and use it then. But this might only be a feeling.

Additionally to that, the card must justify itself. What are you cutting? An economy card? Something else? So I would say, you would need to use it at least twice during a game to make it better then the other card you have skipped for it, and there I see the major issue, as it means you need to see it early to get at least 6 counters. On top of that it also costs you additional clicks to use. That is why I think it's not as good as it seems first glance.

This card creates some mixed feelings and really depends on the breakers you are running. For most rigs it means you save 1 cred per ICE (I ignore Stealth Breaker for this discussion), which means when you get it early enough it will save you the 3 creds you used to install it. But you also need a slot for it, so it should save you another 4 to become a sure gable. Will you break 8 ICE a game? Usually you will, so when you get it early it makes sense. On the other side as it is unique you don't want to have 3 offs in your deck and the other 2 become dead draws. So is it worth the deck slot when you see it late?
On the other hand, when you have rigs that have awkward strength boost like Lustig or Passport it might save you more then one cred on a piece of ICE, main problem is, this are mainly criminal breakers and are currently rarely imported into anarch, while the question is, can criminals include it into their decks?
On the other hand, currently we have the heap breaker (Paperclip, Black Orchestra & MKUltra) which beside of Paperclip have this 3 creds pump strength by 2 & break 2 subs. Especially against Fairchild 3.0 the first pump breaks nothing, similar ICE would be Ichi 1.0, Tollbooth, News Hound and Cortex Lock. So currently Ice Carver shines together with this breakers as for some ICE it will immediately save you 3 creds to break subs and you can see this is a great efficiency and within the anarch faction. So until these breakers rotate Ice Carver is a very strong econ card that can save you dozens of creds during a game. And it has to be seen as a pure econ card that can replace one Liberated Account in terms of value.