Most people prefer desktops, not to say you cant use a laptop it's just easier to have a central station to record.
No means maybe
I'd go with a desktop if you're doing serious recording, like full projects with a bunch of tracks. It'll eat up your laptop's memory and slow it down hardcore.
there was recently a thread about this. the assumption is that a laptop will be more limited, in a large part due to the 5400 RPM hard drive. also, you typically have to pay more to get a similarly spec'ed laptop as opposed to desktop. for any serious recording, i would highly recomend a desktop. if you are more into recording occasionally and just for quickly getting down ideas or something, laptop is fine.
Desktop, since the sound cards are not that good on laptop, also you can get a soundcard that was designed for high quality recording.