How long does a "race" last, and are you actually racing against other people (i.e. start at the same time), or do you just go at your own pace?

Ten minutes n race against whoever's there

10 minute sessions. If you are alone, you'll go alone, but they can have up to 20 people per race. So yes, you are with other people

How fast do the karts go?

There are always other karts on track but the racing is in time attack format. Fastest lap time wins not position

10 minutes. And they regularly send out 20 karts at a time with a slightly staggered start.