Hello! I've just set up/used the Omega drill for the first time since the update. I like the new look! Once I removed it, it left behind a transparent box called "Omega Drill Internal Storage". I can't pick up or remove the box, and I can't place anything in that spot. Why does it do that? What can I do to remove or prevent it? Thank you!

The internal chest is to allow faster unloading via many inserters, as often the previous versions would be bottlenecked by how fast you could unload the connected chest.
The internal chest should always be removed when the drill is. Were you removing the drill in an unusual way?
Edit: 0.3.2 release should allow you to clear orphaned chests like this by opening their GUI.

I'm unsure if it's the cause, but it looks like clicking on my uncommon (and currently operating) drills removes the storage which causes the drill to stop operating until it's re-placed.
If I don't click on the storage though and only hover it, it seems fine.

EDIT: Having started a new save and cheated a bunch of different qualities in, it does appear to be related to the "quality" mod, as the internal storage is not flagged as being the same quality as the drill and remains "common".
EDIT 2: removing a common drill though with it's inventory intact does not yield the contents of the drill either (if I have 250 copper in it, I'll be given none when removing the drill).

Thank you for useful bug report! The clicking-on-storage issue should now be resolved for drills with quality.

Regarding yielding the contents of the drill, at the point in the game where you have these, I typically would prefer to have ore voided than dumped in my inventory or on the ground. (and it's much easier to implement if it's just voided)

Oh wow that was a quick fix, thanks very much!
It works in place as well, so any already placed drills don't break anymore. Although a broken drill will need to be replaced (as the storage doesn't "magically" reappear). Not an issue though, re-placing the drill only takes ~30 seconds, and afterwards it works as expected.

In regards to your second point, agreed, at the point of having access to one of these drills ~300 ore isn't much of a loss.

Thanks again for the quick fix!