What happens once all available rooms on a ship contain a system?
Can a new system replace an old one, or will it become impossible to change systems (except for the medbay and the clone bay)?
No, once you purchase (or otherwise obtain) the last (8th) system available for your ship, you cannot swap out existing systems to make way for the desired one.
Systems in stores will darken and have the tooltip “You’ve reached the System Limit“. Presumably, if there were any events that would grant you a system, it would work like the reactor upgrade and grant you the rest of the award and omit the system. The only exception to this rule is, as you have noted, the medbay/clone bay – which will replace each other.
Also of note; while the Backup Battery is displayed in the “Systems” tab of stores that sell it, it is a Subsystem, and does not count towards your system limit.