Inspection before completion (no stairs and exterior work)

But inside, only the stairs are missing, the rest will be done by Saturday. Outside, yes, it's an absolute construction site, but the cat is not supposed to go outside anyway for the first few weeks. My idea is to move in but have everything open carefully documented with exact deadline dates during the handover. Then they are obliged to complete it by that date, or am I seeing that wrong?

I would not rely on the forum, but on a lawyer.

At that time, we had formally agreed on an acceptance according to [VOB] in the contract. At the handover appointment, we refused acceptance due to numerous defects, which the other party neither acknowledged nor wanted to remedy, and nevertheless moved in afterwards. That we refused acceptance and the reasons for it were recorded in the handover protocol. The other party then argued with tacit (= implied) acceptance, went to court, and failed.
We did not accept tacitly by moving in. The exact procedure had been discussed with a lawyer in advance.

If the relationship is already strained, I would be cautious. In any case, I would secure myself legally and seek advice on how to proceed in your specific case.

Handover protocol

That’s what it’s for: to record defects during the takeover, still accept the house, and then have the defects corrected as noted in the protocol.
But whatever:

If the relationship is already disturbed anyway, I would be cautious.

Me too. In that case, professional advice from a lawyer would certainly be better than a forum :)

That is what it is for: to record the defects during takeover, still accept the house, and then have the defects rectified as noted in the protocol.

If we had only recorded the defects but still accepted, we would have had to pay all partial services and the final payment as if the house had been free of defects. Also, with acceptance, the warranty claim period would have started immediately. The defects were still significant, the general contractor seemed troubled, insolvency was not ruled out. Therefore, the lawyer's advice was "do not accept" in order to withhold money for possible replacement performance. If we had accepted and paid, we probably would have been left with everything, he would have done nothing more, we would have had to sue him and also advance court costs. Through our approach, we have "only" lost nerves, a lot of time and energy. But afterwards, we had a defect-free house and did not lose any money.

Supplement: Surely the recommended procedure also depends on the type/volume of the defects. For minor defects, refusing acceptance might be "using a sledgehammer to crack a nut," but in our case, they were quite large items. Therefore, such a situation can only be compared or applied to one's own case to a limited extent.

What speaks against moving in and at the same time refusing the handover and recording this in the protocol? Or is this legally not possible due to the missing stairs (fire protection....)? The moving company is already booked for the coming Saturday.

Moving in early is simply a risk. You save money on a hotel / vacation apartment / pet care and put yourself in dependence on the fairness of your provider. Officially, you need the approval from the building authority. For the construction partner, this is also an effort and a risk, as they have to secure the construction site differently if someone is living there. We have done the early move without great necessity and moved part of the move forward and partly lived in the house before all work was completed. We were sure that this would work with our craftsmen and got "lucky."