I just got feedback from one of our teams trying to take a test (https://scilympiad.com/stemboost/ES/TQuestions?tid=0000AM). They said syncing would sometimes wipe out already saved answers. They were trying to work on different questions at the same time if that matters but they also said the answer to the first question had disappeared even though they worked on that one together.

Each question is saved individually after the students answer the question. No need to sync the answers. I'll hide the button to be less confusing.

Would each of their corresponding browsers automatically get updated with the saved answer? More generally, what sort of race conditions, if any, can occur if multiple students are answering multiple questions simultaneously?

In the current release, for all question types, except Cryptography, the system can handle multiple questions by partners. The system has been tested with a 40 team tournament without noticing race condition. It's too early in the season and development cycle to have large scale testing.