Does A Teia: IP Recovery meet its trigger condition when Tatu-Bola swaps with an ice from HQ?

Yes.

Can the Corp, playing as A Teia: IP Recovery, use Mitosis to create two new remote servers, then use A Teia: IP Recovery’s ability to install an additional card in the root of or protecting one of these servers?

No. Each instance of installing a card is always resolved as a separate instruction, so A Teia: IP Recovery’s conditional ability can only resolve immediately after the first card is installed with Mitosis. If the Corp chooses to install a card with A Teia: IP Recovery's ability, they will necessarily already have 2 remote servers. It will not be possible to create a new server after this point, so a second card cannot be installed with Mitosis.

What happens if there is more than two remote servers when an effect disabling A Teia: IP Recovery's "Limit 2 remote servers." ability (such as Employee Strike or Direct Access) ends?

The game state is now illegal, so in the next checkpoint after A Teia: IP Recovery's ability becomes active again, the Corp must choose exactly 2 remote servers they will keep. The game trashes all cards in and protecting all other remote servers. This trashing cannot be prevented.

What happens if the previous situation occurs while Architect is rezzed protecting a remote server?

Per its errata, only players and their card abilities are prohibited from trashing a rezzed Architect. Since the game, not the Corp, trashes the cards in the servers the Corp doesn't keep, Architect's ability does not affect it. The Corp is not required to choose a server protected by Architect as a server they keep.