Bathroom redesign extra cheap & washing machine connection

Rip out the carpets, lay laminate. Renew the silicone in the bathroom and replace toilet seats as well as shower heads and taps. Clean the rest very thoroughly. Paint the walls. The washing machine can be connected anywhere there is also a dishwasher. It also works side by side on the same drain. Everything else is nonsense in my opinion. It just has to be clean. Spending money for beauty is wasted.

if you end up at 10k + own contribution in the end and you assume ~one year: does the rental market there also offer something? just as a thought experiment

I am not on site myself yet, but I have spoken with home and the unanimous opinion is that the carpets have to go.

I don’t think it will be 10k of personal contribution.
At most, we will spend a bit more on the kitchen, if at all. We would then move those pieces into the new house. Presumably, the cabinets can then be used in the utility room. And maybe we get a SideBySide fridge instead of a built-in one, since we can then continue using it in the new house.

For the short-term solution for the temporary arrangement, I think we will remove the carpets and put PVC or vinyl or click laminate over it. What would you recommend there?

Otherwise, we won’t do much. Toilets will be partially replaced (but not at our expense), I think I’ll make the rest of the bathrooms cozy with tile paint and bathtub paint. Of course, that is not absolutely necessary. But making the house a bit cozier (gradually, while we are already living there) I count as a hobby. I enjoy that, and you do spend money on other hobbies too. Of course not thousands, but you can achieve a lot with DIY on a small budget.
That was the idea behind this thread as well – whether anyone has good tips or tutorials, or can recommend paints. Less so as a fundamental discussion about our budget. ;)

Lay PVC or vinyl or click laminate over it. What would you recommend?

Definitely laminate. It simply offers the best price/performance ratio if you’re not a phobic of plastics (which I don’t assume given your alternative choices). In my opinion, the selection of very cheap models (<15 EUR/m²) is the largest here, and you can sometimes even get decent stuff for 7 EUR.
Besides, installing it is really doable for anyone (it might be even easier with vinyl, but I haven’t done that yet). For some reason, vinyl is significantly more expensive. I don’t find the feel fundamentally better, nor always suitable for wet rooms (often the carrier board is made of HDF). Experience reports often mention that vinyl gets scratched more easily and can quickly look dull. Unfortunately, laminate is moisture-sensitive if moisture somehow gets between the planks. Vinyl is often praised for its nice soft walking feel. To me, it rather has the vibe of PVC floor covering from the 80s. But that’s a matter of taste.

Vinyl is often praised for its nice soft feel when walking. I think it’s more like PVC carpet from the 80s. But that’s probably a matter of taste.

We currently have vinyl in the new building and I think both are true. :D It is nice and soft, looks good. It is prone to scratches. And barefoot you really get the feeling of walking on 80s PVC carpet. We are still undecided for the new house whether we’ll choose it again. The haptics aren’t the best, but it also has advantages that cannot be denied.

I was still quite vinyl-friendly in the initial phase, but once I realized that it is really just absolutely insanely overpriced (compared to laminate), I somewhat turned into a vinyl-averse person. So if plastic, then laminate. If it is allowed to be more high-quality, then preferably parquet right away (but then solid wood planks as well).