I can't speak to porting over the watch from T-Mobile to Verizon, but I have been using the Gear S3 on T-Mobile (~15/mo) network synced with my Verizon networked phone for almost 2 years now. It works well although it is not as seamless as I think it would be with both on the same network. If I call someone from the watch, it shows with a different number and confuses people. Other than that it works great. I am hoping Verizon gets the new watch soon, but will get the T-mobile version if they don't announce soon.
No, it won't. I tried this week. The salesman thought it would work. Tried calling support for 3 days after purchasing.... finally told it won't work. Also mentions this on the Shop Samsung app in the small print under the Galaxy Watch LTE product page