Shower installed defective + part forgotten during new construction by plumber

Move-in was on 01.06. The plumber forgot an underfloor part on the shower. The shower from Grohe was purchased by us. He agreed and complained. So the finished shower had to be reopened on time by 01.06. 5 weeks later it is still not finished. The missing part is installed and the wall is closed again. Now the shower is leaking. Water is continuously running from the overhead shower and the control. The plumber says it is a design flaw by Grohe. He even wanted to cut something from the shower. Oh man. The plumber has made many other mistakes and is quite uncooperative. We have several other defects with him. What are our rights? Especially since it is still not finished after 4 weeks. We have the mess, the delay, have to take vacation, etc. The plumber apparently does not know the shower well either. I no longer trust the plumber. If I had not been there on Friday, he would have closed the control below with the cover. The water would have flowed into the wall by the liter since he also did not consider it necessary to apply silicone. The gaps were so large that a finger fits between the shower back panel (one part) and the control. That is insane. Can I commission someone from Grohe at the plumber’s expense? For the inspection, special tools are partly required, which the plumber does not have. Currently, we always have to turn off the main valve to keep it dry. The plumber wants to cover the costs for fixing the errors but not grant any further discount despite all the stress.

If the plumber wants to cover the costs, that's already great. Call Grohe, they surely have installers who are more detailed.

Personally, I understand that you want to get a "little more" for your stress. But the plumber's offer to cover the costs is good and not a matter of course, especially when it comes to third-party suppliers.

Regarding the "further" defects you mentioned, one would need to know what they are to comment... In any case, document and address the issues early on. Then, as in the first case, find a joint solution (repair by him or replacement at his expense). PS: Invoice already fully paid?

Classic! He could also always blame material defects.

Your writing is unfortunately very difficult to read. Please list the other defects in bullet points.

The following applies to the material defect: The shower was already installed and under pressure before it was set up again. Everything was tight then. On Friday it started to leak. When I came home, the problem was there. He pressed the cover twice into the fresh silicone and made it completely leaky because everything stuck to it.

Who is responsible for sealing between the fitting and tile/shower back wall? The tiler says the plumber and vice versa.

Further defects
The screed dryer, which was connected to the underfloor heating, was incorrectly set.
Wrong heater delivered, it is larger but the same price, but does not fit optimally in our utility room.
The pipe routing for the heating was planned incorrectly. They run at the back in a niche and the heater in front of it. The niche was meant especially for this. The suggestion came from him. Two weeks before moving in, he says it does not work at all because of maintenance etc.
Pressure reducer in the downstairs toilet not installed. The toilet was also from us.
Pipes in the utility room were leaking.
The heating was defective because a sensor was inserted too tightly. As a result, liters of water ran onto our floor. We were not informed and the water was not absorbed. It therefore remained there for several days under the laid carpet.
The overhead shower was installed crookedly on the first opening from the wall. The wall had to be opened a second time. We had to uncover all errors.
Outdoor sensor was placed incorrectly - south-facing.

That’s what comes to mind immediately. There are more, but for that I need to look at my list.

We had to make sure that everything runs somewhat correctly, but we have no idea ourselves. I am really afraid it won’t work. The plumber acts as if that’s the most normal thing in the world.
The exposed shower downstairs is already installed. I wonder how it was sealed behind it.

The plumber also forgot the plastic parts on both toilets, which actually belong between the toilet and the mounting. That’s not correct either, is it?

When silicone and sealing are mentioned here, I would get an expert opinion or at least a second opinion from a professional. Silicone is a maintenance joint and never a seal. You seem to have already caught a sloppy job here.

Report the defects in writing (very important), set a deadline, and request rectification.