Moisture in the single-family house basement (built in 1936)

Ultimately, the question of whether to demolish & rebuild or renovate depends on what you yourselves want and can afford financially. A house from this year of construction can only be brought to the energy/technical standard of a new building with considerable financial or personal time investment - if this is the desire & requirement, demolition & rebuilding is a sensible and worth considering option. If you are also satisfied with a good, but not "up-to-date" technical standard, i.e. possibly renewing the wiring & heating + possibly the roof + its insulation, renovation is probably cheaper. However, I would definitely not base this decision on the shown pictures of the cellar. Yes, the walls are damp, but as it appears at least in the pictures, not to a worrying extent, but to a desired (!) and normal extent for a cellar of this construction year. As it is now, you definitely cannot make a residential cellar out of the rooms and I would not store paper / organic materials either, but if you can live with these limitations, I would maybe renew the interior plaster (mineral / lime) and leave everything as it is. Excavating + sealing is very expensive and the result is not always the desired one.

The purchase price includes all debts, that is a total of 135,000 EUR.

[*]Basement damp, sealing expensive (we had that) ✓
[*]Basement ceiling height too low for practical use ✓
[*]Wood beam ceilings, with corresponding follow-up problems (repair with beam replacement very expensive, impact sound insulation difficult, tendency to vibration, especially if children's room on the upper floor)
[*]For semi-detached house: party wall with very poor sound insulation. Hardly improvable afterwards - thank God a detached house! :)
[*]Poor floor plans with through rooms - oh yes! A nightmare...

If we were to renovate, we would most likely combine it with an extension. The father-in-law has a drywall company with all kinds of contacts, so we definitely have a big plus in that regard. He tends towards complete demolition. :-D

The issue with the wood beam ceilings might possibly concern you as well. It will be interesting if you want to change the floor plan upstairs or add/change a bathroom, for example. Since the floor plan is obviously also rather suboptimal, you should discuss any ideas for changes with a structural engineer during the planning phase to see if and with what effort they could possibly be realized.
If the desired changes to the house are too large or become too extensive, the option "demolition & new build" would be worth considering. Personally, I would rather invest my money in a house that truly fits my wishes and needs than spend the same amount on a compromise.

If you tear it off, then I'll take the Steuer unit from the oil heating …..:-)

In the meantime, quite a lot has happened, so here is a small update. As can be seen in the photo above, the old floor was broken up and the ceiling height was changed, so that 20 cm were gained here.

This is the room in the first photo after all the plaster was removed and the masonry dried. Insulation will also be added from the outside...