Can a clogged toilet drain cause a tank overflow?
No, a clogged drain won't cause your toilet tank to overflow. Sometimes dirt can get into the fill valve and cause it not to function properly. You could try turning off the water, removing the top  black cover on the fill valve and removing guts and then turning the water back on and flush some water through it ( be sure to put something over the fill valve to prevent water from  shooting out the top) but it's normally much easier to just replace the entire valve. 

As for the water level, in your case you can remove the cover on the over flow tube and cut it down so that it's lower than the flush handle hole. 

Of course all of this shouldn't be done by you since it's a apartment but that would be the way to resolve the issue.