Anyways I'm going to get different RAM, and I can't put in the RAM I want since the heat spreaders wouldn't fit under the heatsink fan, so I was wondering if it should be moved to the other side of the heatsink.
That's exactly it. In cheap cases, there aren't many "cables go poof" compartments, but it surely can be done. When I got serious about it, I rebuilt my entire machine with cable management in mind every step of the way. It took me a good 3 hours, but it's looks purdy.