Hi,
Since you deleted everything from your old Mac and then traded it in, and didn't keep a backup of your iMovie library, I am afraid that at this point nothing can be done. You might have been able to use recovery software had you kept your old Mac, but that option is no longer available unless you can retrieve your old Mac.
You said that you couldn't back up your old Mac to Time Machine "for technical reasons"? Did you have an operating Time Machine backup at one point in time? If so, you could could go back to it and restore your old iMovie library to an earlier point in time and possibly recover some of your projects.
You actually did find your Movies folder and confirmed that your old library is not in it. Your question No. 2 is no longer applicable, but, for information purposes, you would click on your library name in the iMovie sidebar and then do a File/Consolidate Library Media.
Sorry that you lost your projects. That's a bummer for sure. Perhaps you can reconstruct some of them if you still have the original media stored somewhere. This is a hard learned lesson always to keep a back up of your data on a separate drive.
Best,
-- Rich