If I have a garbage bag and I tie it close with some air trapped into it. And I twist the bag in the middle so two bubbles of air appear, did I just 'break air in two'?
No. You simply separated atmospheric molecules into two compartments.
If I use a vacuum seal to remove the air out of my little sandwich bag, is all the air still in that bag? Why then is the surrounding air pushing so hard against the bag that it fits so tightly around my bread?
There will be some air in the bag. If there wasn't then your sandwich would be ruined as the sandwich you came to be fond of.
When you take air from the bag you are adding it to the atmosphere which now becomes part of that atmospheric pressure pushing on the outside of that bag with much less resistance inside of the bag in terms of atmosphere.
It means that the bag will envelope whatever dense material you have inside it.
If a vacuum pump can remove some air from a bag with a sandwich, why can't it remove all air from a glass tube with a feather and a nail in it?
Because a glass tube would simply be crushed, eventually. A bell jar is shaped like a bell because it can resist external pressure safely. A glass tube would be much less effective, meaning a much weaker pump has to be used, which also means much less atmospheric pressure is allowed to expand out of it, which means there will be resistance inside of it, even if it is low compared to external.
Why would there still be atmospheric pressure in a chamber without air?
If there's air in there's atmospheric pressure in. Atmospheric pressure is a mixture of gases that make up air.
Let's make this easier.
Get that sandwich bag and put nothing inside of it. Now squeeze every last drop of pressure from it. What do you have?
You have a container that has become not a container. It ceases to hold atmospheric pressure. It becomes a flat piece of plastic.
Now then, if you do that with a glass cylinder container with a strong enough pump, you will see that cylinder shatter or be breached. Once it's shattered or breached, it ceases to be a glass cylinder container. Why?...Because it cannot contain any pressure. It becomes equalised against it's own wall thickness with atmospheric pressure.
Take a metal container and put a strong enough pump on it. It will eventually be crushed and if it doesn't get breached, it will eventually be crushed flat which then leaves it as nothing more than a sheet of flat metal and not a container at all.
Can you see what I'm saying?