Yes a 3.5" disk drive cleaning kit is universal, it makes no difference on the computers disk file system, nor the density of the drive!
Not sure if you know, the A570 volume knob needs to be turned "on" for it to work. IE like you are turning up the volume on it. You should see a CDTV boot screen with a spinning CD. The games should autoboot if they are a CDTV game.
The first Amiga might already have 1MB of RAM on the motherboard so it will crash if you install the 512K RAM card. Check the RAM on it. I have a Rev 6A board with 1MB RAM soldered on board so the expansion slot cannot be used in this case. The drive sounds dirty or bad. Try cleaning the heads, if that does not work get another drive or swap the other one in it.