if they settle, could the plaintiff sue again claiming he found more damage than he was first aware of?
Generally speaking, no. It would be the plaintiff's responsibility to ensure awareness of what he was settling for. For the settlement to be voidable and be entitled to resume the claims, there would have to be additional circumstances, such as:
- having settled under duress or hardship;
- settlement being induced by fraud (arguably silent fraud suffices for that);
- counterparty's breach of the settlement.
These conditions are no different than the principles of contract law.
Could he claim for something unrelated, like if the renter didn't pay in full the last months rent (assuming that hadn't been brought in first case)?
It ultimately depends on the scope and intent (be it inferred or explicit) that can be ascertained from the settlement. That being said, a catch-all phrase such as "extinguishes all claims" tends to make it harder for the prospective plaintiff to prove that his new claim transcends the scope or intent of that clause or settlement.
I here that in reality this has little affect, as the plaintiff can change something small and say it's a whole knew case. Is it the same for settlement agreements?
What they told you about making "a whole new case" by means of making small changes is inaccurate.
Once the controversy has been settled or become res judicata, making small changes to overcome it is a futile attempt to relitigate matters. This applies to all disputes, whether or not they relate to contract law.
If the changes are not so insignificant, the plaintiff is considered to have waived them anyway for bringing them up only when it is too late.