As said before - before migration you should turn off all external modules and disable all overrides.
Then migrate, then update according to your list (you can update all internal modules at once, no need to be one by one), then turn back on your external modules one by one and observe for problems (blank screen, issues in FO and BO, functionality issues) and then introduce your overrides (if needed with TB).
I think updating in 7.3 should not be an issue (for PS and TB core) but you may encounter issues with modules that are old.
Regarding your restored shop - try clearing all internal caches, the host cache (if present) and your browser cache and check if it solves the problem with the tabs. If you backed up all files by FTP and db is OK this should be the only problem (caching issues).