The idea is "countercurrent exchange": the incoming air and outgoing air travel in opposite directions through the heat exchanger. The result is that most of the heat gets transferred.

At the end of the exchanger, the warmed incoming air is in contact with 70° outgoing air, so it will be at almost 70°. Similarly, the outgoing air will be at almost -10° as it exits.

It will probably make more sense if you look at the diagram: https://en.wikipedia.org/wiki/Countercurrent_exchange

No. Imagine that your heat exchanger is simply two touching pipes running parallel to each other, with flows in opposite directions. Incoming air will gradually warm up as it comes inwards, eventually being warmed by the air which has just left the room. You can get a much better result than just averaging the temperatures.