The schtserv is kind of the definitive private server, so I'm just going to answer the questions based on it. There at the very least used to be more than two private servers, maybe not anymore though, and different servers can vary slightly in content.
Can you still play the Dreamcast versions of PSO and PSOv2?
The server supports every version of PSO except for the Xbox versions as they are integrated through Xbox Live.
Are these servers still online?
The schtserv server is still online.
Can you use your old characters?
Use the commands /save password and /restore password in the lobby chat to save and load your character to/from the server. I believe all character data is stored server-side and not locally, so if you mean your local Dreamcast characters I don't think you can. I'm not entirely sure as to how it works because when I played I'd just re-installed the PC version and didn't have any characters, so I only ever used my character on the server.
If you'd play on several private servers, could you use the same characters for that?
As far as I know, no. This really hinges on where the characters are saved, if it's local or serverside. If they're saved on the server, then you won't be able to.
Can you play together with players on other systems than Dreamcast?
There is PC/Dreamcast and general multiplatform compatability, but I believe it is limited to lobby chat. I don't think you can actually play together.
Are there problems with cheaters?
It's a pretty small community, so I don't think it's much of an issue. I certainly never ran into issues.
Can PAL and NTSC players play together?
You're all playing on the same server, so you should be able to, yes.