This info is already in various threads, but somewhat scattered so I'll summarize:
1: No physical difference. Same hardware on AT&T, T-Mobile, and Verizon units. SIM-Free when it arrives will also be same hardware if Apple does the same as they did with the 5 and 5S. Sprint phones are a slightly different model, but only work on Sprint in the US so I wouldn't touch it if you're not a Sprint customer.
2: Verizon version will work on AT&T, TMobile, or Verizon. Unlocked T-Mobile or unlocked AT&T will work on AT&T or T-Mobile. They'll also technically work on Verizon if you have an activated SIM on an existing Verizon account, but Verizon administrative restrictions prevent you from establishing new service and bringing your own non-Verizon phone. No technical reasons, just policy to not set up a phone that wasn't sold as a Verizon phone.
3: Buy a full price Verizon phone. I suppose you could buy on contract and cancel the contract immediately also, work out the numbers to whichever is best in your favor. Others will need to chime in on this question, I don't consider Verizon an option as their coverage at my home is much worse than AT&T or T-Mobile. NOTE: I don't know how Verizon would treat a Verizon phone that's been off the ranch for a few months.