>it's not pushes-flow-down-so-goes-up.

Yes, it is. It cannot be any other way; momentum is always conserved. You need to get that 9.81m/s^2 acceleration from somewhere!

To answer your two questions:

1) A simple wing is basically a teardrop shaped fairing around a flat plate - not unlike a rudder. Like a rudder, it can redirect the flow in either direction, depending on the angle it meets the air. (In fact, most aircraft struggle to fly upside down - if they're not aerobatic they tend to have wings that are like fairings around curved plates, to more efficiently redirect air in a single direction.)

2) The aircraft in front of you has just imparted a ton of downward momentum to the air. Compared to flight through clean still air, you're on an elevator that's going down.