Calculation of Property Purchase and Renovation

For me, your posting must be divided into two parts.

1. The Financial Aspect
The financing is rock-solid and completely relaxed. Even a fairly high repayment (which is over €500 savings every month) is factored in. So you are basically exchanging your cold rent for the interest costs and your previous savings for repayment.
Everything fits.

2. The Property
What immediately strikes me is the purchase price, which is almost equal to the land value. So you are buying a garden-designed plot for the purchase price. Completely free of charge, you get "a wreck" included. Because the house value is practically stated as €0 in the purchase price.
All alarm bells are ringing for me, because usually no one does that to make a buyer happy. They do it because the building simply has no value anymore and is done for. The risk that you will find problems everywhere is incredibly high.
Before buying, I would spend a week with an appraiser/building expert thoroughly inspecting the house and only then decide. If you move in with a child first, you won’t do proper renovations anymore because you don’t want to live in a construction site for months or stay in a hotel.
With almost €5,000 net (!!!), more than €1,200 rate (also gladly with initially lower repayment) should be possible to directly finance renovation costs.

We spent 2 hours walking through the house with an architect who looked at everything.

The basement was addressed. The steel beams of the ceiling date back to 1900 and are no longer in the best condition. Quote from the architect: "It will certainly last another 10 years, but not 30 if nothing is done about it."
I specifically asked the architect if that is a deal-breaker from his point of view. He said: No, it’s doable, you can still live in it, and it’s not a reason to tear down the whole house.

We would live in the 4 rooms on the upper floor and leave the ground floor as it is for now.

Roof, windows, facade, all no longer new, but livable.

Our alarm bells also went off, among other reasons that’s why we are trying to inform ourselves as best as possible. We have 1 month to decide before a realtor is commissioned to sell.

The house has our absolute dream location. Our entire surroundings are nearby. The property is great and has everything we wish for.

We deliberately chose the €1200 installment to be low. This way there would still be "room" in case the saved money isn’t enough and additional debt needs to be taken on. The interest rate with subsidies for the (energy) renovations for roof/heating/facade could also possibly be better.

I find it hard to bear the thought of paying both repayment and rent at the same time over a longer period; I don’t want this financial double burden to last longer than at most 3-6 months.

Zero if a house has a book value of 0€, it does not have to be junk.

if a house has a book value of €0, it doesn't have to be a wreck.

No, it doesn't have to be. But the likelihood that you will have to do a lot of work on it (that you don't want to do) is very high. Why is it being sold so cheaply? Not to do anyone a favor, but because there is a lot to be done.

We spent 2 hours walking through the house with an , who looked at everything.

Well-meant advice: Don’t spend two hours with an architect going through the house, but six hours with a proper building surveyor. It costs a few euros but is well-spent money.
When I read "steel beam" of the basement ceiling is almost gone, it sends a chill down my spine. Do you know what an effort it will be to replace that? Not just the effort, possibly having to replace the entire basement ceiling and thus also the ground floor, but also the financial burden.

I have a hard time with the idea of paying both repayment and rent at the same time for a longer period; I don’t want to have this financial double burden for more than at most 3-6 months.

You only start repaying after the loan is fully paid out. So in case of financed renovation work, only after moving in. Until then, only interest/provision interest.
By the way, professionals finance the financing costs during the construction phase. You can do that as a private person as well, and then the problems disappear (although with the amounts involved here, I don’t see any real problems – you earn enough for that).

A follow-up financing is always more expensive than the main loan.
Basement from 1900... what gas consumption are you actually expecting? Gas is gradually becoming more expensive. With 8,000 kWh consumption it hardly matters, with 35,000 kWh it really hits hard.

So steel beams have to be replaced? How much does that cost?

And 5,000 net is not as much as you might think. Money slips through your fingers quickly, especially with children, and anyway with food prices.
For your cars, you can safely calculate 700 EUR/month, all included.

And that you move in and really do nothing, just the necessary... I thought that too. Then thousands of EUR in small stuff come along that you didn't consider.

I have a little more than 5k net and since the construction I can barely save anything anymore, on the contrary, I really had to dip into the reserves. [Gartenhaus], [Terrassen(-überdachung)] and [Carport] now have to wait one to three years and I will probably build everything myself. This also has to do with currently low spending discipline, but yes, 5k net is nothing when building a house.