Feasibility financing new construction (land + semi-detached house or semi-detached half)

Presumably because the property would be worth significantly more on the open market.

Maybe that, but it’s relatively irrelevant if you financially fail at building the house. There is plenty of potential, just the earthworks and disposal would probably eat up the €30,000 for construction incidental costs. The landfills in the area are full since a train station is being buried there. Additionally, the presumably missing finishing. Waste disposal, crane, scaffolding are certainly all to be provided by the builder. And that’s how the thousands just tack up.
I also consider the free kitchen a trap. Unfortunately, nowhere it seems to be explained exactly what is included in the kitchen... but the promised Miele appliances absolutely don’t match the cheap, cheap provider Allkauf Haus. Presumably, the money is made again on the back end because the partner kitchen builder massively sells overpriced "upgrades" that are needed to actually make the kitchen usable. Something like that.

And even if the house gets finished, you have to be aware that Allkauf Haus is priced in the lowest league. So you have a cheaply made shell that is then often botched either by the builder himself or poorly coordinated finishing trades. Just the wall construction with polystyrene would absolutely scare me off. Whether €800,000 can really be achieved there, I mark with a question mark.

I don't think much of tax class 3/5, because the income tax return is calculated jointly anyway, so I consider 4/4 to be better and more realistic.

Since the man earns more, but not enough for the loan, tax class 3/5 would be better than 4/4. Whether it then becomes 30/35 or 40 hours is not decisive, but it is a completely different ballpark than 20 hours. Postponing the larger repayment until the children are grown means having American conditions. There, there is no plan to pay off the house at all; instead, in old age a smaller/cheaper property is simply bought and the repayment is the difference between the two houses (minus the new car).

Conditions at first glance... but in the USA there is no fixed interest rate, meaning the interest rate is adjusted quarterly... so if it is at 1.50% at the bank, then everyone can pay nicely... if it then rises within a short time (a few months or 1-2 years) to 3.0% or more, then everyone has a problem.

In Germany, you can neutralize this with the "fixed interest period."

Also, they create value here if the house is worth €200,000 more after completion (current market price) due to the plot price (probably discounted in the region) as well as some personal contribution.

As a counter-question, one can ask: what about the rent in the coming years if we have inflation here?

If they currently had to pay 2.50% interest to the bank, it would probably be below inflation... and the question would be when inflation falls below that?

From these aspects, they would rather build assets (even if they repay little here).

And then they have the option, in 20 years or whenever the children have left home, to sell the house and buy a 2-room apartment. I believe then they would probably already have enough capital for the condominium, and such a goal as: living in paid-off property in retirement.

I don’t know how big their current apartment is. The children are still small, so the demands are probably still low. But in a few years, they will probably need something bigger, then the question would be:
- can they find something?
- what does a bigger apartment cost? €1500 cold? or more? (considering: scarce living space + rising prices!)

No matter how well one works, in the next few years one will have to offset daycare and OGS costs. Taking into account that even a free daycare often consumes about €70 in meal fees per child. With three children and possibly a second car, one would have to calculate what the actual additional income looks like locally. There is more money, which definitely helps and makes sense here, but the stress factor also increases. How this affects possible own contributions to construction is also unclear.

I also consider the free kitchen to be a trap.

You only get the minimum equipment there. As a reference, I recommend taking a look at the online shops of common hardware stores. A fully equipped kitchen unit costs a slim 1000€. Grandma will really be happy to be able to reuse her old stove top covers.

How big is the apartment? If it's too small, is there maybe a bigger alternative through the employer? Because I think you can hardly find cheaper housing in the Stuttgart area.

This one unfortunately got a bit overlooked. The rent is absolutely insane for Stuttgart, so the question is: how was the equity generated, how long has the income/housing situation been like this? For your age, 80k equity is initially great, but with this discrepancy between income and rent, it might even need to be more?! Also, the topic of daycare fees is still missing if mom goes back to work. In principle, everything has already been mentioned; you actually have to strongly advise against it. On the other hand, I understand the situation of seeing a great opportunity here to get a building plot at under 3-400 euros market value. All in all, in my opinion, I see these nasty bait offers here as never sustainable, meaning the house will become significantly more expensive... therefore: not possible.