If the Corp has multiple scored copies of Sisyphus Protocol, can they use both copies to cause the Runner to encounter the same ice multiple times in a single run?

Yes.

If the Runner uses Inversificator to swap a piece of ice to a different position after they encounter it, can the Corp still use Sisyphus Protocol to make the Runner encounter that ice again?

Yes.

If the Runner bypasses a code gate or sentry and derezzes it with Capybara, can the Corp still use Sisyphus Protocol?

The abilities of Capybara and Sisyphus Protocol both become pending at the same time, but Capybara’s ability will resolve first because it is the Runner’s turn. Sisyphus Protocol’s ability does resolve, and the Corp can pay 1 or trash 1 card from HQ, but the Runner will not re-encounter the ice because it is no longer rezzed. Sisyphus Protocol’s ability will not meet its trigger condition on any subsequent encounters that turn.

If a piece of ice is trashed during an encounter, can the Corp use Sisyphus Protocol to make the Runner encounter the trashed ice?

No. If the ice is trashed, the Runner will not pass that piece of ice, and Sisyphus Protocol will not meet its trigger condition. Sisyphus Protocol can still meet its trigger condition later in the turn.

When the Corp uses Sisyphus Protocol, can the Runner use Physarum Entangler to bypass the ice?

Yes. Bypassing a piece of ice during a forced encounter ends the encounter.

If the Corp uses Sisyphus Protocol to make the Runner encounter a piece of ice hosting Kyuban, will the Runner gain 2 after that encounter?

No. Forced encounters with ice never proceed to the “pass ice” step.