You forget, Makarov was an Ultranationalist, who control the Russian government at the time. I'm not going to say the government was likely thrilled about his attack on the airport, but Makarov basically left them with two options: tell the world that a member of their camp led a heinous assault on their own airport, or use Allen as an out. Even if the Russian government knows it really wasn't America's fault, public furor would basically ensure that they had to invade. Hell, it's even possible that they ordered Makarov to do it, wanting a pretext to invade the US (again, these are the Ultranationalists from the first game).
As for the Captain Price thing, he launched a nuke so that the EM pulse would knock out the Russian's advance (nukes are the most efficient man-made way of creating EMP, and a high altitude detonation would leave no fallout). Think about it: they now have no vehicles and are knee-deep in enemy territory. Their defeat would be a matter of when, not if. As for the Russians in the other part of the country, there were none. They only attacked DC and the areas around it, what we see in the brief before the first mission in America was them using the ACS module (more on that below) for a diversion, essentially shutting down the American recon sats.
The ACS module was something from a crashed American recon sat which had been recovered by the Russians. Essentially, it would give access to the American sat network, meaning it had to be recovered (which is what happened in Cliffhanger). However, one of two things happened: Either the Russians copied the module before it was stolen (the original theory put forth when the invasion started) or, more likely, it was just handed straight back to them by Shepherd.
Rojas (the arms dealer) told 141 about the prisoner who Makarov hated in the Gulag. A little anti-climactic, but it still works.
If TvTropes is to be believed, the general consensus is that Raptor is the President. They were quite adamant about protecting him, and it's possible to get a slight glance at him which supports it.
We aren't told who the man in the safe house is, but he was there for two reasons. 1)The men who killed him had tattoos just like the ones Allen had to get before No Russian. Therefore, it's implied that they were Makarov's personal men. 2)There was no sign of forced entry. This hints at Shepherd's involvement.
As for the invasion, I think we can chalk that up to Willing Suspension of Disbelief.
141 were loose ends. They would likely figure out Shepard's involvement from his next moves and, as Price demonstrated in the mission where he launches the nuke, they can get around just fine without him. From his perspective, they were more trouble than they were worth, especially since he had Shadow Company. And yes, it's mentioned in one of the early missions (in passing, but still) that they are his personal troops.
I assume you noticed the large cliffhanger at the end, right? Well, seems to me that Makarov is being set up as the Big Bad for MW3. As for why they go after Shepherd instead of him, his was the more immediate threat, the plan already in motion.
