Well, if you don't want to make an incident but you still want to play, you could create something your character has to do away from home for a while, and come back after you've done that and see if the other person has left.

So is spontaneous RP with a random person rare?

Random RP with random people can vary; it depends on what they're doing, what you're doing, etc. On my server, a lot of RP happens in towns and cities, so running into people and randomly RPing social interaction can be pretty common. Out in the world, it's less likely that you'll run into someone claiming a particular spot at a specific time.

On our server we also tend to have regular "events" like player-run taverns, shops, public IC guild meetings, etc., some of which share physical spaces at different times, that people show up to RP either planned stuff, or random interaction, or both, since those are places and times one is guaranteed to meet others IC and it's acceptable and encouraged to "break into" RP with them without fear of interrupting someone's plot, or even joining into ongoing plot that way.

As for sharing space, it again varies. People can chat OOCly in whispers or whatever to work out a time, or if you show up and find the place occupied, leave and come back later. For instance, if it's something like a private house situation, it's a little awkward to explain Random Strangers joining the RP most times.

In any and all cases, I encourage simple OOC communication in the background to make sure it's OK to join in--and if necessary, how--to keep the RP running smoothly and avoid bad feelings bleeding into OOC. Good RP shouldn't get in the way of Good RP, and a bit of clarifying and communication OOC (don't have to script anything, just make sure people are mostly on the same page) can make IC work a little better, especially when place and time conflict.

"I was just wondering what you do if someone else is claiming "your" spot?"

Ask that nice night elf lady to put some clothes on and threaten to call the city guard if they don't get the **** out of your home.