If someone is bald or suffers from alopecia such that he wears a hair system/piece/wig that is stuck on to his head, can Ghusl and Wudu be performed over such hair systems – whether temporary or permanent? Or must they be removed first, no matter how difficult that may be? The same question also for women with hair extensions/pieces.

They should be removed, as they’ll be a barrier to water getting to scalp (unless water can get to scalp somehow with required amount)
If it’s too difficult then tayamum must take place instead of wudhu