If you take a glance at the description, the Federation Armed Forces base is actually on the outskirts of the city. Given how much havoc would be wreaked, as you pointed out, if they started fighting in the city, you have messed up. Big time. Because it means you've let an enemy get a whole heck of a lot further than they should ever have gotten. The size of the mechs is very much intentional, and was decided with both the technology levels of the game, the military role they occupy, and the nature of this game in mind.
I can address the plot not quite easily as well; it isn't a cold war. The cold war already happened, for the past ten in-universe years. This is the point where the cold war goes hot, and while I am not stating exactly how soon to prevent it from being spoiled, the exact point of change occurs very, very shortly (OOC) after the game starts. It is called "Breaking Point" for a reason.
I can fix the Co-Host note quite quickly, just takes a quick edit.