Can a (20*1.35) tire fit on my rim that has a (20*2.125) tire on it now? Can a (20*1.35) tire fit on my rim that has a (20*2.125) tire on it now? Is there a general rule to know what tires are compatible with the rim?

For information a friend has the exact same bike with (20*1.50) tires on it if it can help...

Thank you in advance
Width compatibility is a matter of rim width, but according to the oft-cited table there's unlikely to be a rim that's particularly good for both 1.35" (35mm) and 2.125" (54 mm).  This is a big change, so unless your rim is on the narrow side for the tyre that's on there at the moment you might have trouble.  So in your case you really need to measure the inner rim width before going shopping.