Do the riders get to practise on the course before the event? It’s a 5km course with 30+ jumps, how do they know where to go?
No. But you've generally jumped practice rounds elsewhere - at home, at other competitions. These are world class riders, they've done many many cross-country events before.
The rider walks the course at the start of the day - generally before spectators etc turn up. You get to judge approaches, and fastest ways through & start to think about speed v risk (eg in cutting corners). You also know how your horse jumps & how nifty they can be in coming to a jump at an angle, rather than straight on, for example, or what they're like jumping water.
I assume the horses that do the dressage aren’t the same horses who do the cross country and show jump. Are the horses they use the riders ‘own’ horse?
In 3 Day eventing or 1 day eventing, yes, you ride the same horse. That's the challenge - you show that the horse is well-trained to your commands in dressage, you show your speed & courage as a team across country, and you show control & stamina in the show jumping.
In some competitions, you don't necessarily have your own horse - if there's a ban on horses travelling internationally, or an onerous quarantine. But I think in Tokyo, all competitors are riding horses they've trained & ridden specifically in world-class eventing competitions.
If a horse was to run around the cross country track on their own, without a rider, would they choose to jump over some of the jumps? Would they run through the water?
Some do! One of my ponies would have kept jumping and then just run off & grazed. He wouldn't go through water on his own, though.
Water jumps are often the biggest hazards for horses. They have a different quality of vision to us (they tend to see light & shade rather than colour) and they don't have great side vision, so spook easily by something at the very limits of their peripheral vision.