Regarding beasts: do they award faction points like heroes do?
Yes. While the exact calculations and other details of Faction Power are unknown, it makes no difference where your military power comes from.
Influence: I thought faction points control how the AI considers your power, the "normal" Influence is soemthign like a currency only wiht no real incentive not to spend, right? Please correct me if i judge Influence, i i hope i don't need to "save" it only to get the max out of all trade treaties all the time
Both Faction Power and Influence have an effect on how the AI sees you, but they work differently. If you click on "Govern" and then select "Foreign Relations", you can see each factions opinion on you and what is effecting it. If you have a certain amount of Influence, you might see a "+1 Your are influential" or, if your Faction Power is low, you might see a "-1 Your military is weak". The bonuses and penalties might rise and fall, depending on your amounts of each, however, there is no penalty for not having any Influence. Ultimately, Influence only improves your relations with other factions, but your military power determines, if they are willing to go to war with you.
If you want the best trades, you need Influence. I don't know if anyone has managed to calculate the exact effect Influence has on trade value, but, in general, it is best to have as much as possible.