Ahhh, Favor. Here's what I know so far...
Demands do not raise favor, fulfilling them, however, prevents a drop.
Requests raise favor, and not fulfilling does not cause a drop.
The effect of gifts wears out with three givings. (*Note, I have no idea if giving the large gift again say.. 21 times will work, I haven't tried it).
Favor generally declines. In the constants.txt file it's 1 point per month.
Finally, as far as I can tell, the old trick of being able to take a lower salary than your rank to gain 1 Favor a month doesn't seem to work.
So... the new trick is to only use gifts in real emergencies (like the Roman Army is coming) or to push you over the top at the end of a scenario.