Hello. Sorry for posting in this old thread but I had almost the exact same problems on a Rev. 6A Amiga 500 motherboard. Only when the memory expansion was plugged in. I tried several different memory expansions that I know works.
One example was when loading a cracked version of The Chaos engine from cold boot. After the cracktro the game stopped loading and most of the time the Amiga would reset and show a guru. If I then clicked the left mouse button and loaded again the game would load up fine. However, if I turned the PSU off and left it off for 5 seconds or more the same guru thing would happen again or game loading would just lock up. I also left Silkworm running in demo mode a whole night and woke up to a guru. Street Rod cracked version would very often crash with a guru after I passed the copy protection by clicking one of the colours/numbers there. Also, when leaving the garage the driving scene would not be there. It would skip directly to the city. And there was something strange about the music.
And I tested this games in my other Amigas where they were completely fine. And again, this only happened if I had a RAM expansion card plugged in.
Yesterday I searched up the mod that can turn the trapdoor memory into chip RAM, to see if that would make any difference. What I discovered was that the chip RAM mod was already done on this board. lol. I then undid the mod and now everything works perfectly fine! Got this Amiga from a friend which had bought it on ebay.
So I thought this was worth mentioning if someone else has similar problems. Just check if the chip RAM mod is used and try and undo it.
Here is the link to show how to mod/unmod trapdoor RAM into chip mem for 6A motherboards