Actually, 'Понятно?' does not mean quite the same as 'Ты меня понимаешь?'. Specifically, there is no "меня" (me) part, explicit or even implied. 'Понятно?'="Is is clear?", it can equally well refer to something I've just said or to a passage in a textbook. In the latter case 'Ты меня понимаешь?' would be simply inapplicable.
It is not that "него" is the "pure" genitive so to call it, which is still "его", just that after (many? all?) prepositions (like "у"), the "н" is added to those pronouns in the genitive (for ease of pronunciation I guess...).
For example, Duolingo has taught us that "нет" requires genitive, so "he is not by me" / "I do not have him" would be (as far as I can tell):
"У меня нет его" or "У меня его нет" (not sure about word order)
Not "него", because there is no preposition acting on "его".
It's not a short version, it's the past tense of perfective verb "понять". As a perfective verb, it does not have a present tense. As a result English "Do you understand me?" can be translated as either "Ты меня понимаешь?", which more literally corresponds to "Are you understanding me" or "Ты меня понял?", which literally corresponds to "Have you understood/did you understand me?".
It's OK to put it behind понимаешь, but in general Russian has a tendency of stressing the last word: https://www.duolingo.com/comment/13955228
So the suggested sentence is more or less equivalent "Do you understand me?" while in your sentence that stress is washed and could even be placed on "me" (although that would likely depend on both the context and the vocal stress).