The scientific evidence until now shows that the Universe is infinite. The space between galaxies has been increasing. This fact is called the expansion of the Universe and was discovered by Hubble, who measured the velocity of galaxies. He found that all distant galaxies were moving away from us, and the further away they were, the faster they receded.

We can think of the Universe like a rising (expanding) loaf of raisin bread. As the bread bakes, the whole thing grows, and all the raisins (galaxies) get further apart from each other. If you are sitting on any raisin, all the other raisins will be getting further away from you. It does not matter which raisin you are on.

Concerning the galaxies in the Universe, the statistics have shown that the distribution of galaxies in space is different from what we first envisaged. Most of them are arrayed in groups, clusters and super-clusters.

Galaxies usually do not occur isolated in space, but normally form groups of several individuals, in a wide range from few or few dozens to large clusters of up to several thousands. If we try to just think about the numbers, yes, there is a high probability of having billions of galaxies in our expanding Universe.

There are lots of ways to look at this question --so I will start with reasonably current theory.

According to the big-bang cosmology model, the universe which contains all of visible space started about 15 billion years ago, about 3x the current age of the sun. If that theory is correct, (and there is substantial evidence) you could only see about 15 billion light years away in any direction since light would not have time to make the trip otherwise. It is not known if the universe is far larger than the one we can observe, but again, there is some evidence that it is. This is certainly not forever, however, and there is only speculation about what happened before or 'outside' since we currently cannot measure it, but there are theories of a an infinite number of slightly different universes, some like our own and some different.

There was a relatively old observation that if the universe was infinitely old and infinitely large, any direction you looked should find a star -- so the night sky should look like the surface of the sun... We don't see that in the big bang theory because the universe is expanding and light hasn't had time to reach us from all points. The expanding big is measured by looking at stars and galaxy spectra and noting that the farther away they are, the faster they are moving away.

As to the second question, there are indeed hundreds of billions of galaxies in the observable universe. You can get an idea, by looking at the Hubble's deep images and by counting-- most of the dots are not stars-- but are galaxies, so far away that you cannot see any detail.