I would agree with all of this!

I would also generally add that horses are very individual, and so a lot will depend on each one's personality and what they're used to. Which means you can get a certain amount of mileage out of "thankfully, X was a very calm horse" or "thankfully, X seemed to be used to bicycles; she swiveled her ears warily but did not freak out as Y had feared she might" or whatever if you want them to be relatively chill about the time and the bike.