The Sat22 has one of the tightest draws of any genesis I've tried, it's been a while since I've tried a Kayfun of any sort but I believe it may be tighter than those as well.
Here's my three answers:
1) I've always used 400 or 500 mesh in my gennies, never really had a problem with wicking.  You'll probably be rebuilding it several times before you settle in on a build you like so it may worth picking up a variety to experiment with.  You'll likely get a pretty wide range of recommendations for this one, and it can depend on which atty you're trying to use.
2) Will a genesis atomizer work with a cotton/rayon/non-mesh wick? Absolutely.  Some people swear by this setup.  Some people will also berate you on here for even suggesting putting cotton in a genesis. 

  I've always used mesh in mine, and a lot of the things I like about this style of atomizer come from the properties of a mesh wick so I've never really wanted to try anything else.  Do whatever you like best, play around with a couple different setups, but I highly recommend taking the time to get a mesh build working properly before writing it off.  It can be frustrating, but I think it pays off big.  We're here to help!
3) I HIGHLY recommend the RSST as a first go at a genesis.  It was the one I started on, and I still dig it out from time to time.  Things like the big open deck, the spring loaded positive connection and the insulated wick hole make things a bit easier when you're first giving it a go.  Also they can be had for pretty cheap which is nice when you aren't sure whether you'll like this style or not.  I have a Sat22 clone, I like it, however it's definitely tougher to setup than the RSST due to the tiny chamber and deck you have to work with.  Haven't used a NextGen but have heard that they can be finicky to get working right.  Another one to consider (though more expensive and a bit smaller work space) is the DID from MMV.  These are probably what you'll find on my mech most days, are pretty bullet proof, and provide a bit better performance than the RSST (IMO).