The Runner uses Baker to run Archives and pays a credit to change the attacked server to HQ or R&D. Does the Corp have an opportunity to rez cards or use paid abilities after the attacked server has changed?

No. Once the Runner approaches the server, there are no more paid ability windows before the end of the run.

The Runner uses Baker to run Archives. If the Corp has a copy of Manegarm Skunkworks, where would it have to be installed to hinder the Runner?

If the Runner changes the attacked server with Baker, Manegarm Skunkworks can only trigger if it is installed in the root of the new server. The attacked server changes before any “approach server” abilities resolve, so if Manegarm Skunkworks is installed in the root of Archives, it will never meet its trigger condition.

The Runner uses Baker run Archives. If there is a Flagship already installed and rezzed in the root of HQ, what happens if the Runner decides to change the attacked server to HQ?

The Runner can change the attacked server to HQ, because Flagship does not have any effect on the approach step. However, Flagship will prevent the run from being declared successful, as well as limiting additional accesses of cards in HQ.

The Runner uses Baker to run Archives. When the run begins, they use “Knickknack” O’Brian ability to trash Baker. When the Runner approaches Archives, can they still pay 1 to change the attacked server?

Yes. The ability that changes the attacked server is part of the overall action ability being resolved. The ability becomes independent of Baker's card as soon as the trigger cost has been paid.