Choose construction financing: What amount/term/monthly savings?

You can of course do that. However, when signing up for a building savings contract, you pay initial and account management fees. If you consider that as an insurance premium against rising interest rates, you can certainly do it. Although the building savings contract probably won’t be eligible for allocation in 2 or 5 years, so you would have to bridge the financing again. We did it because we were also panicking about interest rates (even in the Riester variant...) and we probably lost the interest rate bet. I can’t say for sure yet, we are not that far along in the house planning to finalize the financing. It’s annoying and now we have to see what to do with it.

10k is not equity but a 3-month reserve that one should have as a tenant (at least at your net income) and even more so as an owner.

Building society savings contract? It depends.
If the employer pays VWL allowance, then yes.
At the beginning, only a small building savings contract (5 to 10k). These are not particularly profitable and only live off gifted money. But that’s not bad.
Even afterwards, I wouldn’t do more than 50k BS total, the max is 100k (except for exceptions), otherwise the thing will never be eligible for allocation. And with Riester definitely not.

Next: reduce expenses and increase savings rate (income if possible).

A savings account is mandatory and should be expanded with fixed-term deposit accounts for larger savings projects.
Possibly also 2 savings accounts, one for the house and one for car, vacation, washing machine, etc.

If you manage to save 50k in three years (plus building savings contract) and car, vacation, etc., I see no problem.

Another tip: Unmarried... everyone saves in their own account. It would be stupid to separate that again in case and pay gift tax.

And the most important thing of all: Savings are made by standing order to the savings account after salary credit!
You have already spent the money, at best for the house.
Current account means circulation. My current account has a base amount of 250 € so that I don’t go into overdraft. The rest flows into living expenses, Sparen and Z+T.

In your situation, it doesn't make sense to stress about rising interest rates. You are still too far away from a relevant equity ratio that points towards healthy financing.

Try to keep the savings rate as high as possible. At the beginning, so high that it hurts, making you think twice about whether the handbag or the x-th T-shirt or even the whole grain roll for 80 cents is really necessary.

We did it this way for a full year. As a result, we now know exactly where our pain threshold lies and how much we can maximally make special repayments if my wife and I really put our minds to it.

As Spunk has already written: Transfer the savings rate immediately after receiving your salary to a separate account.

Calmly as written, without any additional buffer for the end of the month, so that just the direct debits are covered + 50€ pocket money.

I promise you, it will work. Then you can build in 2 years and not in 3 years.

At first also so high that it hurts

Laugh... and I thought I was the only one acting like that: You only start saving when it hurts!

I do it exactly like this.
2 savings accounts
1. Special repayment house
2. Savings.

My account hits close to zero already on the 2nd of every month. In the middle of the month, a second salary from the side job comes in and covers the debits that occur from mid-month onwards. I always tell myself that I will blow all the money left at the end of the month, but most of the time even that still goes into savings account number 2.
But I only manage that because I have a specific goal in mind.

I agree with the previous speakers. Ggf. Possibly take out a very small building savings contract, but it doesn’t have to be. Save as much as possible for 3 years! Everyone saves individually in their own savings account first and see what comes of it. If it works out with 2000€/month, the basis is created to start looking for real estate. Saving individually has the advantage that it’s like a small challenge over the next few years and you see whether the partner really wants a property and is willing to make sacrifices for it, or whether after 2 years they prefer a newer car ;-) from my own experience I can only say that it’s worth saving hard for a property for 3-4 years in your situation. A daily allowance account is the right option for this. Don’t let anyone talk you into investment opportunities like stocks or savings plans. At most, a very small building saver for the subsidies. That then serves as a buffer in the first years after purchase/construction.