Here are some answers to your questions: Yes, MathSciNet is behind a paywall. Yes, we ask reviewers to give us copyright to their reviews. Yes, you can post your reviews elsewhere.
In return, for each review, we put US$12 on account at the AMS (up from the US$8 it was a few years ago) for the reviewer to use to buy AMS books, to pay (part of) your AMS membership, or to pay for other things (such as t-shirts). The credit is in the form of "AMS Points".
Why the copyright? This helps us to copyright the whole database. Copyrighting a database is rather different from copyrighting a journal or a conference proceedings volume.
Can you post the reviews? The reviewer letter says, "You may post your reviews on your website, circulate them to colleagues, distribute copies to your students, and make other customary scholarly uses. You may even include your reviews in journals, books, and databases in the particular field(s) of mathematics to which they relate, provided that first publication credit is given to the AMS."
Why the paywall? Creating and maintaining MathSciNet is expensive. It costs millions of dollars per year to run it. The subscription model allows us to cover the costs. It would take a lot of donations or ads to make up for that.
Why so expensive? We work hard to make sure that we get things right. In order to do this, Mathematical Reviews has a staff of 80 people. That includes 18 PhD mathematicians who serve as editors. We also have people with advanced library degrees and experience that ensure that our bibliographic data is complete and correct. We have cataloguers who ensure that we correctly identify authors. We have copy editors who help out with the reviews and check references. We have a whole department who work with the reviewers (and their reviews). We have an IT department.
I hope this is helpful. If not, post a comment!