Rental Agreement - What needs urgent attention?

Yes, I would say so too. Sometimes it can also happen that it says you are obliged to clean the stairwell. Or mow the lawn.

Even if you think everything in the contract is clear to you, I would still go through everything again together with tenant-experienced parents, acquaintances, etc., because they might notice something important.

Pay attention to what is written regarding changes to the apartment. - So, what you are allowed to change and how. The notice period is important. And what happens if you want to move out - do you have to paint, or do you pay a flat rate for the painter... And what extra costs could come your way.

If you want to have a pet, it will probably also be stated in the lease agreement whether and to what extent it is allowed. Above all, check which facilities are supposed to be available according to the lease agreement and whether this corresponds to reality.