A senior professor works at one university, has his own lab and research group(s). Recently, he took another part-time job at a better university. So why do professors need to take part-time jobs?
Of course there's no way to say for sure based on so little information, but I wouldn't assume it has anything to do with finances, actually. The most likely explanation is that he wanted to be associated in some way with the more prestigious university. Maybe just because it sounds impressive (either to society at large or on papers/grants), maybe because he hopes to move there fully in the future, maybe because the association might help him attract strong students or postdocs, maybe because he is collaborating with someone there or managing a joint grant, etc.
Does this professor reserve to right to ask the new faculty to employ or joint pay his research fellow?
There's no way of answering this without much more information. Some people have affiliations with other universities that consist of just a title (with no salary or research funding at all), others genuinely have part-time jobs, and some have negotiated deals involving all sorts of things. In general I wouldn't assume there's any agreement to provide joint funding for research fellows, but there might be.