Construction delay by the developer - Who bears the rental costs?

No. You do have a date stated in the contract

I would first be interested to know whether it is a developer contract (including the broker and developer regulation = you buy land and house as a package from the developer) or a consumer construction contract (you have a company build a house on land that you own). For both contracts, it is basically regulated by law that the developer or the construction company owes you compensation for damages from the contractually agreed date. This claim cannot be dismissed. At most, if there are delays for which the developer or the company is not responsible, one can discuss a postponement of the fixed deadline. However, you would have to be notified that, for whatever reasons, a postponement is occurring.

You can initially claim anything

According to what has been outlined, it is not "anything," but only the legally entitled compensation for damages.

Consumer construction contract:
Building Code 650k para. 3 as a basis for the completion date
==> if a fixed date is stated there, not even a reminder is necessary, as the construction company is automatically in default
Building Code 280 in conjunction with 286 regulates the claim for compensation in this case (no reminder is necessary)

For those interested in reading...
Explicit explanation of the Scientific Service of the German Bundestag within the framework of the 650s.

But: 1. Talk to each other 2. Legal assistance never hurts 3. Submit a claim 4. Reach an agreement and be happy

Sure. He has also guaranteed a handover/completion date in writing.

But only according to the plans and the building description. The clause refers to specific plans.

The construction of the building project must be carried out in accordance with the plans and the building description.

And then came the

regarding "extra requests"

In addition: the client also bears responsibility for their moving planning and financing conditions.

The client here is the developer. The moving planning is not their responsibility.

The client here is the developer.

No, that was corrected. There is no developer.

He is the contractor and we are already the clients as far as I understand. At least we had to take out builders' liability insurance and every brick built immediately becomes our property.

No, that was corrected. There is no developer.

It doesn't make communication with the company any easier either if one is not clear about these fundamental terms.