Just to make this clearer: if you so much as strike an arc on the pressure retaining envelope of a pressure vessel after the hydrotest, you must re-hydrotest. So no, you can't weld insulation support studs or guard mounting clips directly to the interior or exterior of the shell, heads or nozzle necks etc. of a pressure vessel after it has been hydrotested, but before it has been put in service, without having to repeat the hydrotest. But that's not what the OP asked in this case.
You are free to weld to components which are NOT part of the pressure boundary, but which were welded to the pressure boundary before the hydrotest. An example would be a repad, a poison pad, a leg or support bracket, and presumably a lifting lug. That would of course assume that the welding you're doing would be entirely limited to the non-pressure retaining component and would not result in depositing weld material on the pressure retaining boundary.
Yes, the scope of the vessel can terminate in a nozzle, i.e. a piece of pipe, to which piping will later be welded. The weld between the vessel and the piping becomes subject to the piping code and its NDE requirements.
Once the vessel is in service, post-fabrication codes apply and sometimes specific NDE can be performed instead of repeating the hydrotest.