How to move Quiet to a different position

To answer your title - No. I've heard a theory that the reason Big Boss doesn't salute back is because the last person he saluted was The Boss (maybe also why he refuses to salute in the ending cinematic of MGS3)

For the other parts of your question, Punching and tranqing them is all you can do with them - Them thanking you is sincere, it's sparring with them essentially.

And finally, petting DD is something I believe that can only be done out on the field.

You can complete all missions without needing to touch FOBs (except for the mandatory tutorial you mentioned).

But having to go through this does open you up for invasions by other players, and unfortunately there is nothing that can be done about this as you can still be invaded whilst offline.

There is a 'cooldown' period between invasions so you won't be constantly bombarded by enemies.

I'm personally not planning on using FOBs myself in my playthrough - I've been playing since launch and have only been invaded once so far.