There are a few quests that you'll probably want to do to unlock your ability to use things, but if you really want to you can almost completely ignore the quests in the game.

Yes you can but you'll be missing out on a lot of stuff that is locked behind story completion, good stuff

Sure. Some plans are gated behind story quests, but you don't actually have to do them. There's plenty of XP to earn elsewhere.

Some stories just unfold as you adventure Appalachia

You won't be able to launch nukes, get access to vault 79 ect