Does leveling up your coaching staff in career mode change the stats of..

When a positional upgrade has been leveled up it increases a players attributes by 1 point. For example if you upgrade the midfielding staff 1 level a midfield players crossing skill may go up by 1 skill point.

Your players actually do gain skill but it takes time. If you have more stars your player will gain skill quicker. Hope that answers your question.

It all depends on how you want to play the game. If you want an attacking outfit lean more towards your forwards and midfield points. I have been right through a few times now with Plymouth Argyle from 2nd div to European Champs, and I don't bother with finance and scouting at all. A waste of needed points elsewhere.

I don't really know about the coaching staff though I think that it raises the skill level more during a game. I was D.C United in the MLS and I bought Damien Duff. He was 86. Then during his first game it was 83. The next game he was 91. So in a way it does.

Yes it does. If they are good your players will improve faster.