So he thought some quantum radiation must escape.
Wow, simply put, imagine for every particle there is a corresponding particle that is connected. They can differ in spin, but they're always connected --even if they're very far apart. (Hence, "spooky action at a distance.)" Well, what happens when a particle (and particle is a bad way to put it) or "information" falls into a black hole. Hawking theorized that a part might get stuck at the event horizon. Yeah, you're right that, based on probability, some of that information (as radiation) will be reflected (bad term again) back out. This has been confirmed through measurement, but Hawking explained how to look for it.
Anyway, though the amount of radiation is small, it means that black holes can shrink over time. It makes physical sense doesn't it? I mean, if it's absorbing stuff, it's gaining more mass, but it's also leaking. The problem is that no one knows the physics of inside a black hole where there is no time. Um, don't ask me.
Also….I can’t follow why
1. Everything isn’t going into the black hole.
2. The Universe is expanding (they think).
Those things seem at odds.
Why may not be a valid question. 1. Space is big. Black holes form when neutron stars collapse. There are only X number of neutron stars, and none near us, thankfully. 1a. There are super-massive black holes at the core of every galaxy (afawk). That's also been confirmed for our galaxy. So, it's thought that they are actually necessary for galaxies to form. I.e., they are creative forces.
2. Our spiral galaxy, from observation, is expanding and will eventually collide with another galaxy. That can also be measured. Explaining the expansion of the universe is different. Space, itself, is expanding. The common explanation uses a balloon. Just remember that the universe doesn't have an empty core. I.e., imagine the universe painted on the surface of a balloon, and you blow it up. Everything on the surface will move apart. Black holes will also be on that surface. (And, that might affect #1 above).
What's outside the balloon? What's it expanding into? It's not "space" or is it outside space? How do we measure that? What energy is causing the universe expand at an increasing rate. Unknown. They call it "dark," but it's just that we can't see it. But, no worries about black holes sucking everything. Btw, look up vacuum decay.
"A man is rich when he has time and freewill. How he chooses to invest both will determine the return on his investment."