Furthermore I am struggling with my age old nemesis again: inflation is high, in the 20s I believe, but atleast I know the reason, which puts me above my earlier games. Because of my delicate situation with two big threats on either side of my border I have a lot of troops, which makes my military expenses sky rocket.
And I am doing my best to correct this, but it is a slow process that gets interupted when I go to war with the big boys in europe.
Inflation above the mid 10s is bad, bad, bad. It slows down your research, stability regain, increase unit costs, everything. I was crippled as France by inflation of nearly 30% in my first game. Get National Bank idea if you don't have it. Reduce your military spending or disband your armies if you must (if you go to war just recruit them again). And hope for a Master of the Mint advisor.
Should I have annexed Liguria to get the CoT, and possibly enraged France further, or was this decision the superior decision?
It might be better, depending on your infamy.
Am I correct in the assumption that wars where the defeated nation only pays me ducats and gives up cores and releases nations/cancels vassals are ones where my BB goes down?
Your infamy/reputation only goes down with time + any effect of the ruler, modifiers or advisors you have IIRC.
And how exactly does prestige work? I win a lot of wars and usually demand the defeated nation to release countries and pay me and I have a few vassals, but my prestige still floats around the zeroes. I dont think I get the mechanic behind it.
Prestige decays over time (-5% per year IIRC) so that you'd eventually get back to zero in the end unless you have some modifier which slows, stops or (in extreme, unlikely cases) reverses it. Some modifiers/actions that affect prestige are listed here.