No construction start date from the house building company! What can we do?

we have no binding construction date in the contract! Is this legal? Can we possibly challenge it?

I assume that your construction contract is also a "consumer construction contract". Binding information about the completion date must be provided there. If this is not possible, the construction company must provide information about the construction period.
What exactly does the contract say?

We are now unfortunately being stalled. Now the company claims that the production of the house will only BEGIN once the foundation slab of the house is ACCEPTED. We are doing the foundation slab ourselves, but as long as I don't know when the house will arrive, we won't lay the foundation slab which might possibly just weather away??

Now I'll be provocative and claim that you yourselves are to blame that construction is not progressing. Why should I as a company produce a house if you haven't completed your preliminary work?
And honestly. Why isn't the foundation slab already laid? You could have done that long ago and therefore really need to take a look in the mirror. Or will you also later forgo the house because it might also "possibly just weather away"?

So:
1. Talk and read the contract
2. Have the foundation slab poured
3. Involve a construction supervisor simultaneously
4. Wait for the house according to the schedule in the contract
5. Move in and be happy

Such a floor slab does not deteriorate (once it is finished) and nobody steals it either. The risk is close to 0, provided all the plans are correct.

Our house was only put into production after measuring the floor slab.

This is quite normal with our provider (Fullwood).

I would always recommend a building expert as well.

Is that legal? Can we possibly contest it?

Yes, legal. Why contest if the company is stating which conditions must be met? You have talked to the construction company: first the foundation slab, then the house. What speaks against that? So why not just quickly place the foundation slab now? But be careful: Are there any other construction-related requirements you need to fulfill? If you are afraid of weathering, then have the foundation slab covered after acceptance.

For me, the base slab was poured at the end of November 2021 and sat around for 1.5 months.

Yes, ours was from the end of October to the end of February. Nothing happens then.