At 12.5 weeks of age I would give another distemper/parvo shot at 16 weeks. You won't overdose the pup. Like I mentioned earlier, you won't know when the pup's immune system is working and you don't know when it no longer has the antibodies from mom. Mom's antibodies pretty much blocks the vaccine from fully activating the pup's immune system, that's why I recommend another dose.

The neutering won't be an overnight stay. For a young, male pup, the procedure itself will be over in less than 5 minutes. Unless the pup is cryptorchid, then it will take longer, and more costly. Low cost spay/neuter clinics in your area is cheaper than a vet's clinic.

Rabies over here is at 4 months old.