Can we say drive a motorcycle/bicycle?
Interesting question, which reminds me of a time last summer when I was riding a bike. It was my right of way at a crossing, and a car should have slowed down. He didn't, and nearly hit me.
Someone nearby shouted 'Watch where you're driving'. I thought for a moment he was talking to me, but as he said 'driving' I realized he was talking to the man in the car. Had he been shouting at me he would have said 'Watch where you're riding.' And if he had said 'Watch where you're going' he could have been shouting at either of us.
1 Can you take me/give me a ride to work on your motorcycle?
3. In Taiwan, many parents take their kids to school by motorcycle.
But I'm a cyclist (pedal-cycle) not a 'biker' (motor cycle), and I'm not sure whether bikers would be happy with "drive".
2 Can you give me a crossbar/give me a ride to work [on your bike - only with the second option].
The crossbar is the horizontal tube that forms the top of a pedal cycle's frame. "Giving someone a crossbar" means giving them a ride, sitting on the crossbar, without having their feet on the pedals. (This is illegal in the UK.)