The test in the mountains is interesting. For clarity, I suggest you SIP every player and remove all units but a single warrior and a worker, so the human only needs to fortify the warrior, sleep the worker, select future tech, build wealth, and set the research slider to 0% (in that order, the order of your clicks may make a difference in the RNG). Give everyone only Mining, Masonry, and Currency, which shouldn't really make much difference but will reduce the number of things happening so it's easier to notice when the AI changes its moves.
Presumably this will result in history repeating exactly, over and over. Try it a few times to make sure.
Next, try changing things at certain times. Perhaps at turn 10 you change your science build, or increase the slider, or tell the worker to move, or switch tiles worked, etc. Note the changes you make and see if the AI changes its actions in any way when you play differently.
Next, try building the pyramids instead of wealth. See if the AI builds the same as when you build wealth, or if it changes. If the AI builds the same, it means the AI is most likely NOT cheating and the RNG is most likely NOT affected by your build choice. This is not definitive proof BTW. If the AI changes its build, it means either a)the AI is cheating, b)the RNG has been affected, which in turn changes the AI build, or c)something unexpected.
Repeat the 'build Pyramids' test a few times to make sure the AI plays the same each time when you're building the pyramids. Then do the same as you did in the 'build wealth' test, changing the same things you did in that test at the same times, and see how the AI reacts. I suspect that changes which affected the AI in the first test will affect the AI in the second, and changes that did not affect the AI in the first test, again will not affect the AI in the second. At this point we'll have a pretty good idea of what player actions affect the RNG/AI decision making.
Try going wealth -> pyramids at different times, and see how/if the AI reacts to your build changes. Then try going pyramids -> wealth at different times, again seeing how the AI reacts. If there is no reaction at all, no matter when you make the changes, then the 'cheating AI based on your build' hypothesis is disproved. If there is a reaction, AND it follows some logical pattern, then we have a little more evidence to support the 'cheating AI' hypothesis. This will not be definitive proof BTW, it's much easier to disprove things than it is to prove them. If the AI reacts in a seemingly random way, we have a little more evidence to support the 'it's all RNG' hypothesis. Again, not definitive proof.