For example, how would I say "my dad's old car's broken wheel"? Where does the adjective go w.r.t the の particle? 父の古い車の駄目な車輪 ? Is it right for the adjective to appear near の? It sounds odd
It does sound odd, but is correct. Having said that, I think that you could probably avoid saying all that. What might the context of the sentence be? Remember that Japanese expects the listener to use context to help translate. You could break your sentence in two, so you talk about your dad's old car in the first sentence and then talk about the broken wheel in the second.