Can the Runner use No Free Lunch to remove a tag gained from the rezzing of Ping before the Corp can remove that tag with B-1001 to end the run?

No. The Corp has priority when they rez Ping, and its ability resolves in the immediately-following reaction window. After the ability resolves, the Corp receives priority again in the same paid ability window, and they can end the run with B-1001 before the Runner’s next opportunity to use a paid ability.

Can the Runner use No Free Lunch to remove a tag gained from encountering a Funhouse before the Corp can remove that tag with B-1001 to end the run?

Yes. The active player will get priority first in the next paid ability window.