Is there an way to increase the amount of generals in the family (without cheating)?
Bribing, fighting captain led battles, giving your young generals ancillaries that increase children.
The game will try to keep a ratio of generals to provinces (I'm unsure what that ratio is, I've heard 1:1 but I have no solid proof)
You could fight all your battles with captain led armies, but if you already have your ratio of generals; you won't be getting any man of the hour promotions. In my experience there is no factor that affects MOTH other than the ratio of generals to provinces, it doesn't matter what kind of victory you have, how big the battle was, what the odds of the battle were, how many kills you have, how many battles your captain has won. It all comes down to chance and more importantnly how many family members you already have.
So I suppose once you reach that ratio of generals/provinces, no you can't really increase your family other than comings of age and bribing.
On a somewhat related note, I wonder if the ratio of generals/provinces plays any part in determining when an old family member will die. But that is straying away from the topic at hand
Here are a couple general (no pun intended) thoughts and ideas on generals.
If a family member is in his golden years, it might be a good idea to transfer some of his better ancillaries to a younger general. It sucks to have an old family member die and take away all his ancillaries you worked so hard for
You can use crappy family members as delivery men. You load him up with ancillaries to deliver to other family members. For example, that family member governing one of your central provinces doesn't need a bunch of command increasing ancillaries. Likewise with that general fighting armies in the far reaches of the world, he doesn't need a bunch of management points (although influence is useful for when you conquer.) As I said earlier, older generals don't make babies so give the ancillaries that incrase chances of having children to young generals. Basically the idea is not to waste your ancillaries, and you can use those crappy generals to transfer those ancillaries, killing two birds with one stone.
Be careful of what you build. For barbarians taverns have a chance of giving you the slubbdergion (sp?) which sucks. Best thing to do is not build taverns altogether, since their PO bonus is minimal, and completely negated should you get the slubdergion and drunken uncle which I think is also created by the tavern/bardic circle.
