"Can I copy that picture?"
I am very used to "mo" too, and I've heard it countless times from native speakers, unlike this version of DuoLingo that I've never heard or seen. And this makes me quite confident that using "mo" is more natural.
-ていいですか is a way of saying "is it okay to do [verb]?" You can break it down into:
te-form of a verb (in this case, the te-form of する) + いい (good/okay) + です (is) + か (?)
Why no particle after the して? I kind of feel like a は or が should be there. Is this just another pattern I should learn?
It's a bit of a grey area, I think? You CAN nominalize して with something like のことが, which is what you should do normally when you need to use a verb with an adjective. But -ていいですか is an accepted form of it, so it's not necessary.
してのことがいいですか sounds more formal to me, if anything.
その絵コピーしていいですか is marked wrong (and it's not an issue of を being dropped, inclusion of the particle still gets rejected).
Is 絵 not also picture?
It is, but of a different type. 絵 means 'picture' in the sense of a drawing or painting, while 写真 is the photograph kind.
Duo should have used a more specific term if it was only after 写真, though.