Verified answer

Although you did not indicate the depth or width of the trench, it would be exceptionally unlikely that a shallow trench could ever develop into an issue as it relates to the stability or having any effect on your garage structure, foundation, slab or even a driveway. Any water that could or would collect in the trench will peculate out and its cone of influence will be nowhere near your building.

The largest single issue associated with the open trench is a safety issue. Should anyone step in the trench, say after dark if the area is not lighted, they could have a sprain or a break to contend with. You might contact your neighbor and suggest he could reduce his exposure if he would fill in the trench. Make him think you are only trying to protect his interests to get your interests resolved.

10' no not unless your homes where built on landfill and still would be a stretch. Erosion now that's a different thing. Now there's a new low point that has not been there before. First rain look out lol..

Which I guess would ultimately cause undo stresses on your driveway but a lot needs to go first. I'm sure something will be done way before that.

Voted for you Dr Spark On the money ......again Will I ever catch up....LOL