I am very sorry for your situation and your experience, and also about the experience in asking for damages.
The legal situation is as follows: the insurers are indeed not held accountable for such an event. Under French law, they have nothing to do with it as long as you did not have specific insurance for such an event.
The responsible party is the owner of the building. There can be several situations, and since it is not obvious from you question which one applies, I will count them here:
1. If the owners from whom you rent also own the entire building, then they are responsible;
2. If the owners of the garage do not own the rest of the building, the owner of the part from which the flooding came are responsible;
3. Exceptionally, the building managers can be responsible as subsidiary (secondary) defendants, if under the contract they must also upkeep the structure of the building.
In practice, this means you have to sue. It is highly unlikely that anyone would give you damages unless you sue. You would have to sue the owners (option 1 or 2 as the case may be), and you may choose to include the building managers in your complaint. During the trial, you are allowed (and should) ask for an expert assessment of the damages caused to your items. Based on this assessment, the court will allocate damages to you, which then must be paid by the defendants.
This has nothing to do with the rent. You may of course terminate the contract, but you must do so in writing and independently of the damages process. Also, if you do terminate the contract, you should bring in a bailiff or notary public first to assess the status of the items before moving them, as afterwards the job of the assessment judicial expert would be much harder, with the risk of leaving you without any damages at all. Up to the point when you formally end the rental contract, you would still owe rent payments.
You can ask for the health certificate, but it has nothing to do with either of the above. From a legal point of view, it is an administrative law, not a contract law issue.
On the other hand, you are right that they are responsible to give you a proper storage area, and because they have lacked in this responsibility, they should pay damages (unless, as at point 2, someone else owns the part of the building where the flooding started, in which case that person has to pay).
I hope my answer was useful and look forward to your rating, which is essential to my activity.
Dr I L Vlad