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Ok so I am just starting with a 3D printer. I have a RepRap Prusa Mendel with RAMBo board. I am using repeiter-host to control it.

I loaded a 20mm cube for my first print (got the STL off of thingiverse). I heated up the bed and exturuder, then clicked "run job". After seeking home, my printer moved to where it should start printing, then promptly the board ejected itself from the USB. My computer made the sound signaling that the USB got disconnected. I reset the board and reconnected it, but now whenever I try to heat my extruder, the printer disconnects itself again and I'm back to resetting the board and re-plugging-in the usb. Any ideas?

Sounds to me like you've got a crossed wire somewhere. Look for anywhere your wires might be making contact with other wires they shouldn't, or with the board itself. I use a Rostock MAX myself, so I'm not quite sure how the Prusa is set up, but it could also be that your extruder wire is making contact with the hotend, and another wire is as well. Whatever the cause, it seems like when you push current to the hotend, it's going somewhere it's not, and causing the board to reset. The best way to check for this might be a multimeter with continuity testing. Pay close attention to the wires that lead to the hotend and see if they meet with anything else they shouldn't. It could also be that your board is bad, but I would start with a thorough check of the assembly, and a complete rewire if you can't locate the exact problem.

Try this, shut down the printer and computer. Turn the computer on and let it boot fully. Plug in the USB cable and give it a few seconds to register the connection. If you are powering your RAMPS via the PS make sure it it on and completed start up before you plug the USB in. Now start the host software and connect to the printer. What happens?

definetley sounds like a short somewhere, that or your hotend is pulling too much power

Ok so I was gone for a few days. Now when I try to heat my hotend, it does nothing. I tested the hotend heater resistance at the board (with the pins on the bottom) and it was at 4.6 ohms so that rules out the crossed wire. Then I turned the hotend on in the software and it was reading 0.1 volts output. I either have a bad board or some corrupted firmware. Any suggestions?

Ok so after some searching and diagnosing, I found that I have a blown MOSFET for HEAT 1. I want to change the pinout in the pins.h file and use HEAT 2 instead (I will not use dual extruders on this printer anyways). Now I need to know what exactly to change in the pins.h file because I do not want to go blindly changing numbers around and frying my board.
I'm will continue this in a new post as the topic has kind of shifted.