What does the us mean in the sizing? some say 34d and others say 34d us... thanks in advance!

I can’t say for sure, but assume they mean sizing in the USA as opposed to elsewhere.

Are these bras full coverage? I'm a 34DD and need the full coverage

Yesss!! I'm a 38DD & this has to be the best full coverage/full support bra I've had in years! Much better than "VS" or other more expensive ones I've purchased. It feels great! You won't regret it.

Are the molded cups thick enough to provide modesty when it's cold (i.e. will nipples show?)

They are thin and flexible which I personally like. It provides some modesty but if you're worried I would buy some Petals to stick on the inside of the cups.

What size insert fits best inside that won't slide?

Theres not a place to put an insert in these. If you do use any inserts it would be your boobs holding them in place. I would just order which ever cup size you wear. Hope this helped a little.