Which websites pay the answerers for professional advice?

Professional Blogs.

If you want to make money by advising others on a website, you have to make your own blog and charge others access to access it. Jim Cramer does an excellent job of this since he built his rep up at a hedge fund, established his own stock advice show, and has published some books. Because his name is worth so much, some people value his advice enough that they're willing to pay to read more of what he says.

If you don't want to publish a blog, your only alternative is to find a think tank which requires a well built resume anyway. Think tanks also don't stick around forever either since their research isn't always financially guided or grounded, so you have to be ready to jump off as quickly as you jumped on.