The use case you're describing (i.e. using drop paths as an intermediate to publish packages to ProGet) is not uncommon. It's fine, and can be simpler in many cases.
But since you asked... the downsides are that require file share access, new developers may be confused by the process (since the typical workflow is publishing directly), and that there is a lack of immediate feedback mechanism (just because file copy is success doesn't mean package is accepted). I guess those are all obvious.
As to your other questions...
Can ProGet handle situations where multiple computers are trying to install a module from the feed while at the same time that module is being updated by the dropPath feature?
Package versions are immutable and not meant to be overwritten. Instead you are supposed to publish new versions. If your workflow involves continuously overwriting the same version while continuously consuming it, you will get errors. Regardless of using drop paths or publishing.
Could these be potential file locking situations where a package is copied to the drop path and ProGet is trying to access it in the dropPath?
If the file in a drop path is locked for reading (since it's currently being written), ProGet will just try again later.
