"Well then, I'm leaving."
The hover-over hints could use improvement here, especially since they suggest 別れる, which is not accepted, as well as 去る and 置く, and I don't know if any of these are accepted. 置く certainly seems wrong because it is only used as to leave something else in a certain state.
別れる is not accepted because it's to split or separate. It's correct for "leaving", if you say 私別れる
your comment about kanji or small つ？
った tta (next letter twice type. here "t") /
っ ltu (this means "little tu")
Is this literally just "I'll go and come" or "I'll go and then come back"? Like, 行って、来ます?
Yeah, it is. When you have a verb in te form, and then you add another verb in its present form, it means I'll and then.
I put あの for well then, which I guess is wrong, and it corrected it to では - what even is that?
It is not '(行っ)できます' but '行ってきます'. It is not 'de' but 'te'. It is one of the greetings when use go out. :D