The connection issues are not uncommon - You likely have either a brownout (power draw causing a dropped USB connection) or a ground noise issue. We have a few common causes listed here: How to troubleshoot connection issues with USB
C3D should also be able to help on their forum.
Regarding the ‘runs off board area’ and starting issues, it sounds like your motors are skipping really bad when you run a job. You never show a pic of both what you tried to burn in the UI and how it looked on the laser when it was done, so I can’t tell if things are sized right, but I suspect they aren’t. If your K40 isn’t bog stock, you might need to adjust the settings in the config file on the C3D’s SD card for your hardware. If the machine is moving farther than it thinks it is, it also means it’s moving faster (or trying to) which could be the bulk of your problems.
The ragged edge on the image engrave is most likely caused by trying to run an image engraving too fast (more than 80mm/sec using the stock firmware with default settings). That’s fixable too, but let’s start with the first bit.