What is a probability of a particle launched from earth hitting any object in the universe? Imagine I pointed a laser into the night sky. How likely is it that a particular photon will ever hit anything?

This question bothered me for a while. I know very little about astronomy. When I asked my roommate he laughed and said that it must be very little, since the universe is mostly empty. I know it is true, but it seems to me that volume of the objects in the universe does not matter. Rather, it should be the area of objects when projected on a sphere that matters. 

On one hand, universe is pretty empty, so hitting something outside the Solar System might be hard. But on the other hand, there are a lot of possible targets. These targets, however, appear smaller when the distance increases.

Which one of these, or other, forces will dominate? Would it make a difference if we ignored gravity?
I can't give you an exact number, but the chances your photon will hit anything within the next several billion years is exceedingly small. Nay, infinitesimally small. Almost no chance that it will happen.

What you're really asking, stated as a physics question is, what is the mean free path of a photon in the present universe?

If we consider an object (be it an electron, photon, or baseball) traveling through some medium (be it the Sun, space, or Earth's atmosphere), we can calculate how far, on average, it can travel before hitting something. This distance is known as the mean free path. You want to know how far a photon can travel through our universe before it hits something, in other words, what is the photon's mean free path?

The answer is that a photon's mean free path in our universe is larger than the observable universe. What that means is your photon can travel from Earth to the edge of the observable universe and still not hit anything (on average).

To support this fact, I point you to the Cosmic Microwave Background (CMB) radiation. This is radiation which was produced more than 13 billion years ago in the early universe and has been traveling through the universe since, almost completely unimpeded. In fact, if you tally up all the photons in the universe, you'll find that a very large fraction are CMB photons. This tells you that these photons, despite being produced 13 billion years ago and traveling through the universe since then, have never hit anything since and most are still traveling through space, unimpeded.

You are correct when you say that its the projected area of objects on our sky that matters, but you have to take into account the extremely minuscule projected area the matter in the rest of the universe has, primarily due to how far away it is.