Can anyone tell me what aspects of the Sandbox instance stay the same after a refresh? Also, is there a way to keep Sandbox integration records after refreshing? We have an API that uses Sandbox for testing and would prefer to not have to create a new integration record each time.

A refresh is a complete new snapshot from a source account into a bare environment, so nothing changed in sandbox since the last refresh will be retained. You'll almost certainly have to create a new integration record, unless you can either mirror it in production, or have two sandboxes, refreshing one from the other (only an option on Premium and higher tiers).