Why space expansion affects matter?
Space actually has energy, vacuum energy. It has been shown by various experiments and can be explained by Quantum Mechanics. So more space means more energy. Although it probably can't be used to do work, it does act to increase expansion. Check out the wikipedia page for more info.
Although galaxies are massive, they are far away and thus the resulting acceleration towards each other is weak. If the space between galaxies is expanding at a faster rate than their mutual atraction; galaxies will move apart.
Imagine a football player running from one endzone toward the other as fast as he could. However the distance between them endzones doubles every 4 seconds. The endzones are not moving; the space between them is growing. There is no way the player could hope to reach his goal. In a short time he would not be able to see the other end.
The scary thing is this not only goes for the field but everything. The player himself would be ripped apart as his own body parts moved farther and farther apart. If space was expanding fast enough the attractive forces keeping black holes or even atoms together would not be enough. This what is commonly referred to as the Big Rip.
Luckily the expansion is slow enough that only "distant" galaxies are moving away from each other. Gravitational forces in a solar system or a galaxy are more than enough to withstand expansion. The larger the scale of the system the more expansion has to play.
When we talk about expansion we are not talking about objects moving away because of their great velocities against a static backdrop.
What you're describing is like marbles on a grid. The marbles get farther away due to their velocities relative to the grid pointing in opposite directions.
Instead expansion is like the grid growing in scale. This effects the distances between the marbles independently of their velocities; which is why we observe all far away objects as moving away from us. If it wasn't for expansion we would expect a more random distribution. It is only at smaller scales where expansion is less of a factor that we can see objects moving toward us such as Andromeda's Galaxy.
We are not in any sense moving away from the center of the universe. The idea of the Big Bang and Inflation is that not only every thing, matter and energy; but everywhere was contained in the Big Bang.
Space is "nothing"
Not really. General relativity tells us that space-time is an actual living entity that responds to what's matter doing and matter responds to the way space-time is curved. Or as John Wheeler put it: "matter tells Spacetime how to curve, and Spacetime tells matter how to move." For an introduction into general relativity, you can start at this wikipedia article.
Gravity doesn't have a cutoff distance
You are talking just classical Newtonian gravity. But this isn't how gravity works on big scales. On big scales gravitation is just an appearance caused by the fact that the space-time is curved. If it is curved in a certain way then the appearance is that the objects attract each other. Like this:
But on huge scales these local, attractive, deformations are completely negligible. When you take whole space-time into account, it is curved differently (more onto this in the last section). And the important fact is that the space-time expands (like when you blow into a balloon). Because it expands, the distances between any two points are getting bigger and it seems like everything is getting away from you. So that gravitation is actually a repulsive force on big scales (or at least, it can be; again see the last section for a clarification).
Wouldn't the expansion decelerate?
There are many solutions of Eistein's equations. Some of them are stationary, some of them are expanding (either at accelerated rate or decelerated rate), some of them revert from expansion to retraction. Now, which of these solutions is correct depends on the precise conditions at (or more precisely shortly after) the big bang.
Important part of the present theoretical understanding of the space-time expansion is that the cosmological constant is non-zero and this is what accelerates the expansion. The complete picture about the matter content of our universe and its relation to the expansion of the universe is called Lambda-CDM (Lambda is an alias for the cosmological constant and CDM is a cold dark matter, which makes up most of the matter in the universe). Now, cosmological constant is also called dark energy. But currently nothing much is known about these matters and major conceptual problems in modern theoretical physics and cosmology have to do with the nature of the cosmological constant.