grammar - Order of words and punctuation in a sentence

I don't think the sentence is excessively long, but you could reduce the ambiguity by moving one of its elements to the front:

After the show ends, ask him what becomes of the dogs he claims to have rehabilitated on the show and he leaves them again in the hands of their ignorant owners.

You could also simplify the "leaves them in the hands" part:

After the show ends, ask him what becomes of the dogs he claims to have rehabilitated on the show and they {go back / are returned} to their ignorant owners.

If, as Sven Yargs suggests, the emphasis of "after the show ends" is on what happens when the pets are returned to their owners, rather than on the timing of the question to be put to the show's star, it might be better to break out the long preceding phrase that defines which dogs the speaker is talking about:

Ask him what becomes of the dogs — the ones he claims to have rehabilitated on the show — after the show ends and he {leaves them in the hands of / returns them to} their ignorant owners.