Q: Is it possible to hook Blaise CATI to our own telephone solution?

Yes. An API-example is shipped with the system, called ServerEventsListenService.

Q: Is it possible to restrict which projects are visible on the server to specific users in CATI?

User must be granted rights for Server Parks; so this allows to determine which projects are visible for the user.

Q: Is there a way to exclude cases (other-mode only, ongoing or finished in another mode) from the current and future Daybatches in CATI?

Yes, you can handle this in the rules. Make sure that the questions are off-route if a value has been set (e.g. WebComplete := 1)

Gaining deeper understanding