MacBook Pro repeatedly freezes, often with odd display glitches
The symptoms you describe could all be explained by a failing GPU system but before seeking a hardware repair (or quote), I would try reinstalling a clean version of OS X - perhaps the original version that shipped with that Mac to ensure that the hotness and slowness isn't because of added load the new OS has provided and actually something relating to hardware cooling or heat generation.

Your picture is very telling as well.

The three things that hardware could fail are:


LCD panel and cabling
GPU itself
VRAM


The patterns you show in single user mode could either be software or hardware. If it's hardware, they will get worse over time (perhaps dramatically worse) and we've already covered the software angle. Usually when the hardware is faulty, you can get poor graphics in single user mode or the boot selection screen - so you might be right it's overheating / the hardware starting to fail.