Sometimes, with no events or other likely causes, the scores change to absurd-looking values.

For example, my king dies and his heir has a prestige of only around 200, so lots of vassals start declaring war, as they are wont to do. After successfully besieging a few of them and telling them to revassalize, suddenly the revolt intensify, and I look -- prestige jumped from about 200, with an increase rate of 1 per month, to -20K. What?

After a long and bloody civil war, the situation improved and vassals are loyal once again -- but that's largely because prestige somehow climbed to 56966. What?

At this points the values don't mean anything anymore.

I've had the same kind of unexplainable yoyo going on with gold -- once jumping from around 5600 with a monthly income of 200 to suddenly -3000. Again, no event of any kind seemed to be triggering these drastic changes, it's just a new month starts and instead of adding 200 gold it removes nearly 9000. And the income is still around 200 gold.

Piety seems to be behaving correctly so far though.

Another thing that looks like a bug is that armies sometimes spontaneously disappear after a victory. Again, I'm not talking about events causing a regiment to disband, and I'm not talking about an army routed out by a crippling defeat. They win, they had minimal losses, and they cease to exist. The regiments can be re-recruited at full strength instantly in their home city.

Anyway all these oddities make the game a lot less enjoyable than it could be.

Well, there are explanations for the issues you observed with prestige. When making peace, pay close attention to the "war score" that is depicted at the very top of the negotiating screen. Depending on the claims your opponent has and how thoroughly you thrashed him, the score can be pretty high (in the thousands is likely if they got a claim on a kingship title of yours when revolting). Once you set your conditions, the game calculates a seperate "peace score" depending on your conditions.
Assuming you forcefully took control of all fiefs, the war score seems to end up at a number around what you would need in order to resolve all claims in your favor (as your opponent not longer has any powerbase and is at your total mercy). However, if you do not resolve everything in you favor (because you want to keep them as your vasall) it is possible that the peace score is much lower than the war score, or even negative.
If the peace score is lower than the war score yet still positive, you end up getting the difference in points as prestige, presumeably because you were merciful where you had no reason to be. If the peace score is negative, though (which is likely to happen in the scenario if the revolter has a lot of titles including ducal ones yet you recognized their titles while not forcing them to step away from their claims on you), then you strangely loose prestige according to what the peace score is, ignoring the war score altogether (because you are unreasonably forgiving, I guess). Similarly you would loose prestige if the peace score is higher than the war score somehow (can happen if part of the fiefs of the revolter are conquered by another entity, yet they have to accept whatever deal you present because they lost control of all their fiefs during the war), here you would loose the difference between peace score and war score, presumably because you forced larger concession out of them that you had any right to.

So those issues are kind of weird, but consistent when you pay attention to the peace terms and the scores associated with them. While you can "game" the system here a lot and bring your prestige to insane levels you can also loose a lot for no obvious reason if you pay no attention. So I would believe what you observered there is "as intended".

Regarding the money: With the change in prestige it is possible that parts of you court went to become disloyal. While the obvious cause would be members of your staff being disloyal, I guess you would have figured that out. Sometimes weird and very bad events can play out in the background even if your main staff is still devotedly loyal to you (had my treasury ransacked at somepoint recently because my character's wife was extremely disloyal yet friends with the the person in charge of things. However, you would still get notified of the outcome of these events with an event card. They are sometimes easy to accidentally click away, especially if lots come in at the same time or you were trying to give orders of something. Have not seen a change of gold without any reason at all at 180+ hours. It is possible that the event cards might not appear if you specifically changed the options to do that, but is should still appear as such in the log at the bottom of the screen.
I also have no clear explanation for your armies disbanding on their own. Disloyal staff would be the main reason (again, influential friend might also be the cause even if the staff person themselves is loyal), but you should still get an event card in that case.