You are correct. When people say that the probability of a random event approaches 1 over time, it's usually under the assumption that probability is constant. For flipping a coin, assuming a constant probability for "heads" every flip is reasonable. For a person, assuming a constant rate of train-death every year is not reasonable because, as you point out, various things will affect your probability of getting killed by a train, such as "whether you're alive," that change over time.

>Over an infinite period of time the probability of the universe ending is 1, right? Does that not invalidate every mathematical assumption that the probability of [any event] occurring over an infinite span of time is 1.

The first question is a physics question that we don't really have an answer to. Even if it's true, that just means that there will almost certainly be a finite lifespan of the universe, meaning there's no infinite time span for the relevant event anyway. More to the point, math, including probability, will always be at best an approximation of reality. You generally can't assume you'll actually run into infinite anything, but there are situations in which you have so much of something (such as trials of a random event) that you can expect it to behave more or less like the infinite case (such as giving a very, very high probability of eventual success), so doing math about infinite cases can still sometimes serve a practical purpose.

You're right that you can think of gravity as something that isn't a force, but it works differently than what you describe in your post.

Essentially, an object with no (non-gravity) forces on it will trace out a straight line through space-time. This is basically inertia: an object won't change speed unless some force acts on it, and if you chart out the path an object moving at a constant speed traces, it follows a straight line. Technically, it's called a "geodesic," not a straight line, but that's essentially the general concept that includes "straight lines."

Gravity happen because geodesics tend to get closer to massive objects. This seems strange; for instance, if you were on a spaceship moving next to the earth at exactly the same speed in exactly the same direction, the paths you and the earth are taking would be parallel. There'd be no significant forces affecting your ship and the earth, so you'd expect those forces to stay parallel since you're moving in a straight line, but your spaceship would get closer and closer to the earth until you eventually crash. This is because spacetime isn't "flat," and geometry in curved spaces behaves differently than geometry in flat ones. As an example, if you take a globe and draw two lines next to each other, perpendicular to the equator, those lines look parallel. But if you continue to draw the lines straight (technically, drawing geodesics on a sphere), they would intersect at the north and south poles. That's because the surface of a globe is curved in a way that "parallel" lines eventually cross.

Something similar happens with spacetime. Mass (and energy) itself causes spacetime to curve. It doesn't seem to curve into anything, in the same way that a globe curves in 3d space, it just sort of intrinsically is curved. We also don't really know why energy curves space, beyond "it just does." But it does curve spacetime, and it curves it in such a way that the geodesics that other objects follow tend to get closer to it over time. Heavier things curve spacetime more, which makes geodesics get closer in a more pronounced way, which makes objects fall faster.

Why would the pressure crush you? There's not really anything compressible in the human body besides air pockets, is there? Isn't the biggest problem with high-pressure diving breathing gas?

We do know the precise values of √2 and 1/3. They are, well, √2 and 1/3. We can also calculate their decimal representations as accurately as we need to as well. The fact that their decimal representations are infinite is more a consequence of how we write numbers down than anything else. For instance, if we used a based 3 number system instead of base 10, 1/3 would be written as exactly 0.1.