Q #1, You'll have to take it out of the bag and put it in a plastic container (which is a crock of BS in my world...) but they make ya do it.
2. You can carry another small bag sure, as long as it does not exceed two, I believe. I don't fly WN (rather get a kick in the balls than fly more than 2 hrs with them) so I really don't know... Southwest forum can tell you for sure though.
Thing is... unless you're really small, or the laptop is really small, using one in coach is not comfortable by any stretch and I've read a few posts where reclining people have snapped monitors... probably won't happen but just an fyi.