Question

1. He is riding a horse now.
2. He is riding on a horse now.
(Which one is commonly used? Do we omit 'on' frequently?)

3. He is riding/taking a bike.
4. He is riding/taking a bus.
5. He is riding/taking a car.
6. He is riding/taking a train.
7. He is riding/taking a plane.
(Are they all the same? Are 'riding' and 'taking' interchangable in the sentences?)

Answer 1

We would more commonly say "he is riding a horse", not "on" a horse, but both are correct.

"riding" and "taking" are interchangeable, yes. We commonly say someone is riding IN a car, "taking" a train or a plane. We might say someone is riding IN a plane or a car, but it's correct the way you have phrased these sentences.

If one is riding a car, I have an image of one straddling the roof as one might straddle a horse or a bike. One rides in the car. The same with a train or plane. We do say we "take" the train, bus, or plane since they are public conveyances, so we "use" them or "take" them.