Gravitational forces do indeed weaken with the square of the distance, but they strengthen between more massive objects. You also need to factor in the relative velocity between the objects.

Just for a complete picture, the Andromeda galaxy is about 2.5 million light years away from the milky way (~0.9 Megaparcecs), and moving towards us at about 110 km/s. The expansion rate of the universe is about 73km/s per megaparcec.

While this is probably a massive oversimplification, we can think of the Andromeda galaxy receding from us at about 73 * 0.9 = 65km/s, but is moving towards us faster than it's receding, at 175km/s, but the expansion rate of the universe brings its net speed down to 110km/s. This is just a snapshot in time, since as the distance gets smaller, the force of gravity gets stronger, and there's less expansion of the universe to deal with. In other words, Andromeda should be accelerating towards us over time, assuming just gravitational force and the expansion rate of the universe.

Why didn't Andromeda and the Milky way already merge? Basically, both galaxies were originally two separate denser regions in space that were far enough apart to form two independent galaxies, but close enough to not recede over time thanks to the expansion of the universe.

Ever bit your tongue before? If yes, then the nerves in both your tongue and teeth are not 100% effective at preventing you from biting your tongue. Now take away all of your teeth's nerves, and you're going to increase the likelihood of biting your tongue with less nerves.