my dad's old car's broken wheel
I think it's okay, grammatically speaking at least.
You could use 壊れた for "broken", though, like:
父の壊れた古い車の車輪 would normally be interpreted as "wheels of my dad's broken old car".
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 ...
As @AeonAkechiさん hinted at in his/her comment to the OP,
is the same as
One of the things we like to do in colloquial Japanese is leave out particles like を, が and は (in their neutral forms), and even sometimes に. It can almost always be assumed from context what particles are missing, but it can be ...
This 'の' should most naturally be regarded as an apposition, rather than possession. So it refers to a 姉さん, who is your 親戚.
Looking up a dictionary, the definition of the word 姉さん usually starts with these two:
older sister (広辞苑: >「あね」の軽い尊敬語…)
young lady (広辞苑: > 若い女性を呼ぶ称。)
In the phrase 親戚の姉さん, it falls somewhat in between. I think "a female, comparative ...