#1
I'm going to sound like a total noob now, I've actually been playing for about 5 years, I just haven't investigated classical guitar that much.

Firstly, what's the name of this style of guitar(where the guitar is lay facing up on the lap)?

Such as, http://www.youtube.com/watch?v=6KrU1LY-pQU&feature=related

Secondly, can anyone recommend any simplish songs in this style which I could learn as a beginner?
#2
This is the normal position for playing a "Dobro" style resonator guitar, and of course, as this guy is doing, you don't need a dobro or resonator...
Normally, this is done with a slide and played in open tuning. The technique is a little different than "bottleneck" slide, you normally hold the slide in your thumb and index finger and mute the strings with your little finger.
Look up "lap steel" or "dobro" on YouTube.