A few questions to go through, but here we go!
Continue working on your scores...try to maintain an average of at least 4,000 before jumping into cash games. Also try to fluke a score of 8,000+ every once in a while, whether it be through hypers or lucky chains.
I dunno anything about Hangmania, I think Gamergoddess knows the scoring standards better.
Try working on your BJ scores for a week or two before plunging into cash games...I suppose it's OK if you're just using the bonus money.
I would put in $20 to get a $20 match plus the extra rewards points. Just withdraw your $20 back.
Yes, start with the cheap stuff until you feel a little bit more comfortable wagering in higher stakes games. But if you find yourself playing particularly well against a ranking group, take a plunge at some higher stakes stuff ($2.50 / $3.65 entry games).
Pull out your wins to however you wish...I always try to pull out when I make a profit for the night, and then start fresh my next cash gaming session. 
You CAN challenge friends for cash.
For Rewards Points, you'll be able to check a list of prizes along with the points needed to redeem something. It can be as little as $5 in game credits (which is really easy to accrue points for) or up to a $500 Visa gift certificate.
There ARE 2nd, 3rd, 4th, 5th, 10th, etc. prizes on both WW and King. There are different tournament formats with their respective prize and placing structure. You'll be able to check them all out once you make a deposit.
Jacob, aka Meikyousisui on King.com & Worldwinner.com