enemy military power skyrockets

Looks like they either bought some mercenaries or were gifted some ships. You have Mercenaries installed? Check the market roster to see if they grabbed any.

The AI has a small chance of getting free tech and money (among other things) on higher difficulties. This is most likely where you see the 5K out of nowhere.

They also get free Ideology points, so it is likely that they used Ideology to unlock traits that grant free ships.

Sharp rises in power and rapid drops in cash can also come from upgrading a fleet.  I am not sure if the AI can do upgrades. It would explain this though.

I did it once using the fleet commands page to upgrade all all of one type of ship.  Big Mistake.  I went from a barely growing 50k in the bank to over (under) -200k credits.  There was no check to see whether you had enough money to do it.  That one action tanked my plans for about the next 100 turns until I got back in the black.  Fortunately my new fleet was strong enough to hold off the baying wolves.

No, Mercenaries is not installed.

Is it possible to pin this down using the cheat mode? After all, this is the third time this happens to me and we now have several hypothesis what could have happened.

You will not see the money grant using God Mode, but it is in the game's code. The AI is set to have a fixed chance to receive grants at regular intervals, which, IIRC, is around every 10 turns. That is not all, however.... If you look closely at the timelines, you can get a good idea what happened....

If you look closely, you will notice that on turn 50, the Yor has a sharp drop in their funds to match the sharp rise in their military power. The most likely, and probably only, explanation for this is the production of ships because ships are the only things that will sharply raise your military power at the cost of economical power. All other methods of raising your military power will not be as drastic (even though I am saying drastic, a rise of like 30-40pts is not much lol) and will not impact your economy negatively.

First, you should know that you are allowed to go into debt, but once you are in debt, you are forced to change all of your production to maximizing income until you are out of debt. The same works for the AI. This is why you see a sharp rise in economic power on turn 52, right after the sharp drop. The sharp rise on turn 54 could be due to the AI grant kicking in, among other things (like decomming ships and finding artifacts/pods).

Secondly, there are mainly two ways that ships can bankrupt your economy so suddenly: free grants (e.g. ideology) and simultaneous build completions of many expensive ships. If you see that the Yor suddenly has more ships, then that is probably it. However, if you look at the ship construction graph, the Yor had been steady up until that point in ship construction, so it is unlikely that they had a bunch of ships compete their build at the same time. If they were bankrupted by multiple ships completing their construction, you should have seen a sharper rise in ship construction as well. When they ran out of money, their ship construction crashed as well until they were out of the negative.

Other possibilities are less likely to explain the sharp drop all the way into the negative since you cannot actively spend more than the cash you have on hand. Only maintenance costs can drive you into the negative. Considering all of that, my bet is that they use they used Ideology grants--among other things-- to give themselves a bunch of free ships, which was much more than their economy could handle, and thus crashed their economy. Their military power did not dipped despite being in debt, so they did not decom the new ships to make up the difference. Instead, they probably just got away with free money and changing their production.