This will be difficult The following comments are in order of your paragraphs above...
0. General comments.
The Andromeda Galaxy is one of the nearest galaxies to our Milky Way Galaxy, and on a dark and moonless night that has no clouds, where you are away from the light pollution of small cities, you can see it with your naked eye, and easily see it in binoculars or a small telescope at fairly low power. It is so large (several degrees wide (the moon is 1/2 degree wide in the sky), so lots of power will not help you see it) that it does not fit in the field of view with my larger telescope.
1. How can we predict or know
Spectroscopes identify the velocity of stars and galaxies. Each element has a known set of spectral lines, and these can be identified and then the velocity can be determined using the principle of Doppler shift. The Andromeda Galaxy is one of only a very few galaxies that are approaching us -- most are moving away. Astronomers call it "Blue Shift" when moving closer, and "Red Shift" when moving away. Anyway, the radial velocity between the Milky Way and Andromeda Galaxies is known quite precisely.
Astronomer Edwin Hubble discovered that there were galaxies outside of the Milky Way, back in the 1920s and 1930s. Certain stars in the galaxy were identified as belonging to a special class of star whose absolute brightness can be determined, and from this, the apparent brightness allows distance to be estimated.
2. Light at end of tunnel of the Universe
We've not been able to detect an end (or edge or outside boundary) of the Universe.
3. Tell what the Andromeda Galaxy correctly looks like...
We have the technology. Telescopes on the ground, and the observatories in space have taken marvelous photographs of the Andromeda Galaxy. Have you seen the pictures other galaxies in the Hubble Deep Field and the Hubble Ultra Deep Field? Those pictures are awesome, and show thousands of galaxies in places that previous observations seemed to indicate were "empty" space.
Remember you CAN see the Andromeda Galaxy with your naked eyes (and better with binoculars), so taking pictures of it is easy.
4. But how do we know for real
My answer above about your first paragraph tells us how we know for real -- astronomers have measured the Doppler shift of the spectroscopic lines from the Andromeda Galaxy and have found they were shifted up in frequency (toward shorter wavelengths, or more "blue") so we know for real that the Milky Way and the Andromeda Galaxies are approaching each other. Some fairly precise information is included in the Wikipedia article I've referenced below.
5. If the Andromeda Galaxy is heading towards us...
But, it IS heading towards us. There are plenty of other galaxies which are actually "crashing into each other" but in fact, most of the space in a galaxy is empty, and some seem to just pass right through each other, sometimes merging, sometimes just passing through. So, when a galaxy hits another galaxy, large parts of it do not "stay there" because they are pretty transparent.
As an experiment, set up two fans to blow their stream of air at a common point a few feet from each other, where the air streams "collide" at nearly right angles. Now, move around outside the collision area and see if the air streams come right through each other. (some will). There may also be a "merged" stream that contains some of the stream from each fan.) Air is much more dense than galaxies are, so you can see that a lot of the stars and planets in a galaxy are likely to be light years from the nearest thing they could have actually collided with.
So, it is more like airplanes flying past each other because they are on different tracks through the air, and so not colliding. The car analogy could be close if there is very little traffic. Think about the drunk driver who gets on the road going the wrong way... he may be able to drive a long way before having a collision, especially in early morning hours.
6. You still need good replys
But if you actually read what's in this, in the Wikipedia article, and in the other articles it references, you'll find really good answers, perhaps instead of just replies.
Andromeda is not likely colliding into other galaxies, since if they were near it, we'd see them too. Given how empty space is, it is probable that the Andromeda Galaxy has a totally clear empty path until it comes close to the Milky Way Galaxy.
7. Think about it...
As you can see from reading this and the references, a lot of people who are astronomers, astrophysicists, and other space scientists have given this a lot of thought.