If you don't have one, get a metronome. They are the greatest aid there is in learning rhythm and building up speed. Like everyone else said, start slow. Up the speed of the metronome by increments of about 4 or 5. You'll build up speed before you know it. This is how I learned Flight of the Bumblebee. I stated with a tempo of around 120 to learn the notes, then increased the speed by 5 bpm, got comfortable at that tempo, then increased another 5, and so on until I finally hit it at 230 bpm. This took me about 2 weeks.

As for ear training, there are a lot of different ways to go about it...(by the way, perfect pitch is rare). Anyway, try humming or singing along to what you play. Practice a scale and hum along to it. This is how I improvise when jamming, except I "hum" in my head and play what I'm "humming." I picked that up from listening to Jimi Hendrix. He was one of the rare players that would play to what he was singing rather than sing to what he was playing.

As for sweep picking, I'm still trying to figure it out. I can do 'em, just not cleanly. John Petrucci's Rock Discipline video has a decent section on sweep picking.