Yay for getting the motors and endstops going.
If there is too much friction in the x-carriage movement at any point in its travel, the x-motor will stall.
If you take the x-belt off, you should be able to move the x-carriage by hand fully from side to side without much variation in the force required. If not, you've got alignment issues with your parallel rods, and you should fix that before worrying about motors, electronics or mounting the extruder. And yes, with the x-belt removed the x-motor should work fine, because it will be completely unloaded.
Ouch about the melted bit. This is one reason I suggested getting the firmware set up before playing with motors or heat. You want to know your software system can report temperature accurately.
In both sprinter and marlin firmware, there is a lockout that stops the extruder heating at all if the thermistor temperature reading is less than 5 degrees. That way, if the thermistor comes un-connected for some reason (evil crimps, I hate them) nothing gets melted. Regardless, don't even >try< to extrude if your printer control software doesn't report ambient temperature within a degree or two (Celsius, of course).
Using an unmodified version of Marlin on my machine gave me totally crazy temp readings - like 127 degrees as ambient temp. I didn't try to extrude, as who knows what the real nozzle temp would get to.
The three most likely reasons I can think of for your thermistor to read zero are:
1. The connectors are open circuit or shorted together. You say the R is good, so it sounds like you've already tested this. If you hadn't, I'd suggest that you unplug the thermistor connector from the ramps board and measure the resistance across the connectors leading to the thermistor. If you get no connection, something isn't plugged in (usually a crimp isn't making contact). If you get zero ohms, you've got a short between the two wires, or the leads of the thermistor itself. Without knowing what sort of thermistor you've got, I can't guess what a good resistance would be.
But it sounds like you've eliminated that possibility, leaving
2. You've plugged the thermistor into the wrong place on the RAMPS board. It needs to go on the pins labelled T0.
and
3. The firmware is expecting the thermistor to be connected somewhere it isn't (meaning you've got the wrong board selected in your firmware configuration file), or the firmware is expecting a different sort of thermistor (also a change you'd have to make in the config file, though I'd expect silly numbers rather than exactly zero degrees if this was the case).
If you haven't flashed your arduino with your own choice of firmware, who knows what it is currently set up for. And if you ever >had< succeeded in flashing a test program, that would have overwritten whatever software was already on the arduino (some people don't seem to understand this).
I still think you should get your firmware working before playing with motors or extruder any more. Especially since you can't extrude anyway til you get your replacement part. And even more especially since you >can< get the thermistor working without trying to heat the hot-end, or extrude.
Once you can flash your arduino with your firmware (sprinter, please, to begin with!), and got meaningful ambient temperature readings back to pronterface (or something less trustworthy* if you must), then you can play with the endstop settings if you need to, and calibrate the x, y and z movement distances.
Then you could strap a pencil to the end of your extruder and practice using the printer as a 2D plotter! Or occupy yourself slicing .STL files into Gcode with Slic3r, or designing things in openscad or blender or whatever while you wait for your replacement part to arrive.
Best of luck,
Andrew Roberts
brazenartifice@gmail.com
brazenartifice.wordpress.com
*by trustworthy, I mean something >I< trust and understand. You may have good reasons for using something else, especially if you are on a Mac.