It just adds base weapon damage to his melee attacks.

No, only auto-attacks.

Any non-ability damage. It will mitigate towers (including Rook's towers), minions, and auto-attacks, but not any ability (Snipe, Fireball, Spit, Bite, Pounce, so on so forth).

Edit: Missed one

You mean those bonuses that show up on items? They are just item bonuses, so while assassins can equip them, they don't get those bonuses, they're extra for Generals. In response to the Rook, its shoulders and spawned towers do not count as minions.

Like the green + signs on his skill tree at the bottom.

Yeah, those "stat" skills are actually unique to each Demigod. They are generally the similar thing across each Demigod type (assassins and generals), just with different numbers. So Assassins basically get all of what JagerJack listed, and Generals tend to get actual minion bonuses in them too.

I don't think most people take the stat upgrades. There are usually enough good skills to sink them in. The stat upgrades are good if you take all of them, but that's a lot of points which are more precious than an item slot.

Pure Frost TB also uses the stat ups. Rook has ALOT of really useful skills and passives so he tends not to get them. 

for generals the Morale (minion boost) passives are pretty common choices for minion builds of Oak or Erebus. Sedna and Queen probably not so much.