1. For FCPX you will want an OpenCL compatible card. Look Here:
http://support.apple.com/kb/HT4664
Basically, the gpu helps to accelerate your playback and previews. But, I believe with the Macbook Air you are stuck with the Intel HD 4000 which isn't listed.
2. Minimum specs all depends on what sort of files you are editing. HDSLR files are usually quite small and low data rate, so you don't need quite as much, compared to DVCPRO HD or some AVCHD. But generally a 4-core cpu over 2 GHz will get you by, and at least 6-8 GB or RAM. But RAM depends on what you are doing. Doesn't sound like you are rockin' the AfterEffects, so 24 GB of RAM may be overkill.
3. Certainly. Many people edit with their laptops. Once again it all comes down to what sort of files you are dealing with.