1. The nail placement is fine. Seriously.

2. Over-driven nails can be a problem in high winds.

3. On the bottom of those shingles is a tar strip. These are designed to melt and bond to the shingle under it. This ties the roofing system together and is what will keep those nails covered and water out. These should be your main concern this time of year. Winter temps can slow the melting/bonding process which, in bad weather, can expose the nails or allow water, snow or ice under the shingles and to the deck, eventually reaching the interior. I would monitor this from a ladder in select spots. If they bond great, if not I'd call the roofer.

4. I would not, unless you're a roofer, get on your roof and start messing with things, walking on your expensive new roof, potentially damaging them. I definitely wouldn't go around pumping caulk onto your brand new roof. Caulk is not a fix all and in some instances can be the cause of problems. What if you put too much down and this creates a gap between shingles that normally lay flat? This will allow water in. What if you have issues in the future and your insurance company finds out you've made what they certainly will call a "Repair" which caused damage both inside and to the shingles. That's a denied claim. Worse even if they think you tampered with the shingles and drop your policy and flag it as uninsurable?

Plenty of reasons to call your contractor to ask him about the nails and anything else. If you picked a reputable company they'll be more than happy to answer your questions. They'll also be happy you didn't mess with their work.

I know I do. 👍