1. Influence. For normal factions, you already got everything - you get it from buildings, and it influences trades in your favor.
But some factions are different - Altar can get influence from quests, and spend it on henchmen, Karavox can turn units and cities, some nations can recruit mercenaries. I guess it should be explained, but it should be explained in the details of the specific strategies that use it.
2. Occupational penalties start at 50%. They stay at 50% for a while - I would say about 10 turns. Then they start decreasing, and go down to at least 20%. I honestly do not know, because usually my games do not last all that long. In any case, you are better off razing anything smaller than level 2, and keep anything bigger. On higher difficulties, conquered cities are way better than your native ones, even with the occupation penalties. When the AI has unlimited money, significant growth bonuses, and seems to start with 4/4/3 default capitals, you want to keep these cities.
3. Upgrades are messed up, and the developers seem to consider that low priority. Check this old, old thread. The only thing that got fixed was the exploit. The bugs are all still around.
By the way, 'best available' seems to only take into account the 'main' damage. I.e. the best spear is the one with the best piercing value.
4. Dense champions - no, this is kind of low, unless it scales down by the number of nations in the game. For comparison, in my last posted playthrough (large map, 8 nations, ridiculous/ridiculous) I ended the game on turn 149 with five heroes (plus sovereign) and there were two more on the map.
Those were 1L1 (starting one), 2L3, 2L5, and the one I did not pick were L5 and L9. The AIs did not get these because I am very aggressive about closing my borders.
By the way, wilderness armies can kill heroes and other monsters.