MiL sounds great to me.
Making driving lessons conditional on good behaviours sounds reasonable for a barely earning adult. It is reasonable & fair, considering that getting a driving licence when still a teen is a great boost to independence & opportunities. Also, it confirms that being on the road as a driver is not a right but a considerable responsibility.
Interesting that she put conditions on up front to get her funding & endorsement.
Which leads me to OH’s father. Is he around? What kind of father was he to OH? Reasonable ? Good example to follow? Did he sew the seed of good fathering when OH was young & consistently into teens?
It seems to that OH has missed some fathering somewhere along the line & does not have positive experiences, or role models, to draw upon. If the cousin is paternal, how is your OH’s relationship with his siblings?
Do they reflect upon their upbringing & subsequent parenting styles?
Is your OH clear about what makes for good parenting & working together to raise children? Is he clued up or drifting, maybe doing it differently/better from his point of view?
I wonder if MiL was the parent who did the tough stuff by choice or the father was absent?
All of which leads me to suggesting that talking to your GP or Relate for some solo/couple/family therapy may help you & OH open this issue up so that you both & your kids can live together better. All families have fissures & pressure points, & there nothing quite like the teenage years to pile the pressure on to expose any flaws.
The 15yo shoving the much smaller child is not a good sign. It is not an acceptable expression of frustration by any measures. This needs to addressed so that LOs are safe & secure in their own home. Taking time out to be with 15yo is a good start.