Maybe.

Determining which possibilities this is false for in our particular universe would take some time, and depends on the exact form of the laws of physics (which we don't know), so let's use a simplified example.

Take the Game of Life. While simple, it is in fact turing-complete; this was demonstrated by implementing a turing machine on it, which is the best way to demonstrate that sort of thing. (It's fun to put one cell out of place and watch it disintegrate.)

Take an infinitely large game of life. Start it in a random state, and leave it to evolve for an infinite amount of time. As you'd expect, a lot of things happen; in a universe that large, you will indeed see, for example, all possible simulations of Earth. So in that sense, "all things happen"...

But there are some states of the world which you will never see, no matter how long you wait. These are called Garden of Eden states.

There's a very good chance that there are also garden of eden states for this universe. They're likely to be states such as "The universe is tiled with a mandelbrot pattern of black holes"; states which are simply so unstable that they cannot naturally arise. There may also be less exotic states of that sort, but I feel less secure about claiming that...

And the Garden of Eden theorem, if it is applicable to our universe, states that it has Garden of Eden states of and only if time is non-reversible. As physics does indeed appear to be time-reversible, that's a bit of a problem. However, I don't know how applicable it is to our non-cellular physics.

Not necessarily. It pretty much depends on dynamics laws of the universe: you can have states that are consistently and forever missed. There are theorems for the eternal returns in both general relativity and quantum physics, that elucidate these questions.

Isn't it pretty established that the universe is not infinite?

In any case, I don't think so. Even in an infinite universe there is the possibility of loops or repetitions. Also you can have an infinite but not comprehensive set of events even if those events are all unique.