The EGR lowers pumping losses. It's essentially a vacuum leak minus the O2. So, ya, it raises manifold pressure (less vacuum). It also allows for greater throttle opening angle which results in even less pumping losses on the intake stroke.

Help push the piston down? Semantics. 360view is correct on this, the wording is just...unconventional. I don't know about the heat transfer thing.

You ever hit the highway in a throttle cabled car w/EGR and notice a few minutes into the drive that you have to push the accelerator pedal down a little further to maintain speed? That's when the EGR opened.

2-4 mpg in about normal.