Over the winter my wife and I are going to be building 2 semi-auto UZIs, I'm already aware of the blocking bar, notch in the bolt, etc. for the SA build.

Is there any issues with using the rear section of the original receiver with the original serial number and markings? Really like to do this with both builds if possible, assuming I can find a saw cut receiver rear half that's still decent. I've built many 80% kits and always used my own name and serial in the build. Trying to go original here.

I don't know those legalities but here's video of Mark Serbu re-building a cut Uzi.

100% no issues at all......

If you're concerned about having your makers info on the gun, or decide to go the SBR route, engrave your info on the bottom of the receiver.

Rear chunks with enough "meat" on them are getting tough to find.......

This.

You don't need any markings on a Title I firearm (unless your state/locality requires them).
To make a SBR (Title II) just add maker's name and city, state of the build. You can adopt the other original markings from your stub.