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.