Okay, let's talk about those things. Oh, wait, we already have.
Is there "physics" in the game world? Well, there's the physics of the game world, which is to say that there are some aspects that are acknowledged already in the rules.
So if I, as a DM, say, "You fell 30 feet, so you take...", and a players complains, "Wait, you're trying to bring physics into a magic oriented game world", I cast a huge, double-empowered Dispel BS on them before they finish the sentence. Things like falling damage, water pressure, air pressure etc. are already accounted for in the rules. (See Falling Damage, the section of Overland Travel relating to climbing high mountains, and spells like Cloak of the Manta.)
So, is there water pressure? Already established. Will water flow into an open and unprotected container if you immerse it?
Sorry, but that last question was so stupid that I threw up in my mouth a little while typing it. We shouldn't have to even ask it, but some people don't like the answer and try to invent excuses to question the obvious. (In fact, I have a bet with myself that someone will try to argue it, even now. "What if it's upside down? Why would it work across dimensional boundaries. Are you sure there's air in the game world? What if it's a Tuesday and I have a Royal Fizbin?"", etc. )
Now, to more serious questions: Can you use the bag as an air supply? Yes. the rules say so.
Why do all bags have the same amount of air? I rationalized an explanation, but the real, according to RAW, unarguable reason is because the rules say so. Come up with the in-game explanation of your choice.
Will air pressure change the amount of air in the bag, and thus change the weight of the bag? Since the rules say that the bag doesn't change weight when you add things, I'd say it doesn't change weight when you add things. Even something as dense and massive as air. Of course, the air in your game world may be more massive than the air in my game world, so you'd better check with your DM.
Will a bag sink? By the rules, a person with 15 pounds of gear will sink if they fail a swim check. The lightest/smallest bag weighs 15 pounds all the time, and can't make a swim check.
Will the amount of air inside make a difference? No more than the amount of gold or anything else. We could get into air displacement v fresh water v salt water, but since the bag changes neither size nor weight whether full or empty, it doesn't matter. The inside is a non-dimensional space (not actually a separate dimension, just a space that takes no space), so whether it's air or lead, it doesn't displace any water on the outside.
Will it sink bottom-first or top first? Since the bag's weight is a constant, no matter what you put in, or how you distribute it, I'd say... Ask your DM. In the real world a sealed bag that's heavy enough to sink does so based on where the weight settles. Since it can't settle in the Bag of Holding... Ask your DM.
If we make the bag invisible, can you see out of it? That's actually a really good question. Since your head is still ijn the same dimension (bags are non-dimensional, not extra-dimensional), the answer should be yes. But because of the spacial distortion that comes from having your head take up no space, the answer is really anybody's guess. And by that I mean, it's the DM's call.
Now, what happens if the portal to the non-dimemsional space closes while something is part way through. I'd say, see Gate. Whatever the rules say there is what I'd apply.
Does the weight of your hands, with or without heavy gauntlets, count towards the total weight of things in the bag?
How much do your hands weigh? If you're that close to the limit, and have actually tracked the weight of every coin, pebble, wineskin and fleck of dust in there so you know that you're right at the limit... then you have far too much time on your hands.
Technically, they should count, but I can't imagine staying in the game with any DM so predatory as to pull that on you.