Can he a) look for a job there and work?

- Yes. He can be on a family visit visa under your sponsorship. Once he gets a job here and a company can give him a work visa, he will transfer sponsorship.

B) can he apply for a job from the UK and be under his own sponsorship and I can be under mine??

- Yes of course. Now, it all depends on the job offer he will get from the company here.