[Kaslak]

I think I can contribute only a little to the first part.
In the answer I am assuming a sort of serious commitment from the characters, as if it was a recognized combat sport, just for illustrative purposes.

If you want to limit character progression around a developement story(perhaps balance time spent on basic education with the combat game training) with no hard boundaries, you might ban "stress based learning", (get points per session and spend them) and use training rules. There are good options (incl. Training workshop sessions) , and an hybrid system to reward points per session to boost training speed in social engineering: back to school. This would move the development of the characters away from session to session scale into the downtime scale (weeks, months, semesters).

Also finding good quality instructors and the money to pay them could create interesting noncombat quests, and the need to balance out the characters related to real life skill needs may be a soft push for players not to focus too much on the sport. Maybe some characters are opposedd by the parents and need to use real life skills to get out from home to train, or may need to work part time because they are struggling. Maybe, as you mentioned, solving mysteries with other life skills might be the only way to prevent a cheating opponent to win automatically.

On the other hand, if your target is to put the campaign spotlight only on the combat game itself, I might expect that characters will want some options to advance on that field only, and any deeper character definition would be a nuisance. The mecha customization options (with point costs) may absorb that need. I have read sometime ago on the forums, to make things interesting, you might create dedicated techniques for different actions based on driving, to favor distributing points on different things and have specialized pilots...