How do we know that this was the actual beginning of the universe?
We don’t. It is “the beginning” in that it is the farthest back the laws of physics can be “rewound” before they break down (in the sense that once you rewind the clock back far enough, there’s enough matter and energy in a small enough space that the interactions between quantum mechanics and gravity become increasingly relevant, and we just don’t know how those play together; see Quantum gravity)
We know that space is expanding faster than light can travel. How do we know that the Universe isn’t trillions of years old, but we just can’t ever see it because it’s beyond the distance that the faintest detectable light can travel?
We don’t. But there’s no evidence to suggest that the universe outside of what we can see might be different from the parts we do see. We can’t really speak much about what’s outside of our light cone (what we can see). All we can say is that the parts we do see all looks basically the same (homogeneous, astronomically speaking).
This is the “big crunch” question, and I think the current theory is that no, there’s enough energy, and space is curved in such a way that it won’t crunch back down. At least not all at once? We do have black holes that are sort of similar in that they pack so much stuff into small spaces that the current laws of physics are unable to completely describe them.