Is handing Save the Dates/invites/thank Yous out in person rude?

Not rude at all as long as, like you mentioned, everyone present is invited.

Ditto.

But it also might make it a little more complicated for you. Like, if someone you think will be at the family event isn't there that day so you tuck their invite away in your purse and then forget about it. Or your recipients leave their invitations at the party and forget about it/it gets thrown out. Or people start giving you verbal RSVPs on the spot and you don't have a way to keep track of the yeses and nos.

Sure, stamps seem like just one more expense to cut, but if mailing everything out at once simplifies your life, it can be a worthwhile expense.

I don't think it's rude. I've received them in person from our co-workers, but on a personal level I like the idea of sending them out. People still like to receive snail mail

It's not rude unless it's done in front of people who are not also included.

I don't think it's rude as long as it's not done at an event where other people are there who aren't invited.

I think this is perfectly fine, but I probably wouldn't do it if you've having a formal wedding. If your wedding isn't especially formal, then I'd say go for it!

Not rude but I think it makes it less formal to hand it to someone (more the invitations than anything)

I don't think it's rude. In our culture, my Mom emphasized that I have to hand deliver the invitations to our relatives. According to her, it's a form of respect.