The set of full imports/full syncs will happen even if the server is in staging mode when you have first set up the server. Switching from staging to live will not trigger a new round of full import/syncs.
Password Hash Sync runs separately from the import/sync/export cycles, and it does not run while a server is in staging mode, meaning that if a server goes from staging -> live for the first time there is a catch-up period for it to go through the entire set of in scope users. Some would refer to this as a "Full Password Hash Sync" - as it is going through all users rather than just watching for changes and processing the delta. This is luckily pretty quick unless you have millions of users being synced via AAD Connect, so I would expect the entire catch-up process to take less than an hour. It's probably worth doing this on an off-peak time just to ensure that the period where Password Hash Sync is catching up for the first time doesn't cause any trouble, but that's about the only "gotcha" you'll have here.