So, the problem is in fact of a hardware nature. By replacing RAMPS 1.4, the problem went away.
Arduino, firmware, G-code... everything is the same, only RAMPS was replaced.
I'm going to find a suitable MOSFET replacement.
My first thought is heat creep. I've never seen a heatsink on the hot-end that couldn't use more air and a lower temperature. IF not heat creep, then perahps under-temperature.
But, since you've worked on this issue without success, and the extruder skipping happens even wit a steady flow and no backpressure against the bed or part, perhaps we should look ...
Sounds like you have a short in your sensor or sensor circuitry. However, if this is the case the sensed temperature is higher than the actual hotend temperature. Does your hot end go cold when the sensed temperature takes off? A short could also show the hotend temperature going up at an unrealistically fast rate. This doesn't seem to be your case.
Using a larger carriage will, if too large, diminish the printing area.
To determine the correct values (for the X-axis) you:
First need to assure that the nozzle can reach the origin, if not, try to see if you can move the endstop without the carriage hitting the frame, if not possible, your left margin will be decreased. If there was enough ...