Is it possible to install a washbasin in the toilet room afterward?

Status. Toilet removed.
Pretext installation with wall-hung toilet.
I think then you have a better starting position / options.

But the drain pipe is vertical in the floor. We don't want to break anything open.

Without a basement and without any work, it will probably be rather difficult.

I would do a pre-wall installation similar to this, but with the new toilet placed at a 45° angle in the corner and then mirrored:



Then everything disappears into the pre-wall; if you extend it all the way up with a built-in cabinet, towels and so on can also fit in the upper half. Cover the wall with the basin completely with mirrors and everything appears bigger – the toilet is positioned at the corner and you don’t have to see yourself in the mirror while taking a crap ;o) Bright lighting or on one side a fake window with lighting. Use large format tiles or cast material on the floor and walls and suddenly the whole toilet has the "wow" factor.

How deep is the room, where is the door, how wide is it, and does it open inward or outward?

I would do a pre-wall installation similar to this, only so that the new toilet is placed at a 45° angle in the corner and then mirrored:



Then everything disappears in the pre-wall; if you pull it all the way up with a built-in cabinet, towels etc. also fit in the upper half. Cover the wall with the basin completely with mirrors and everything looks bigger – the toilet is located around the corner and you don’t have to see yourself in the mirror while pooping ;o) Bright light with that, or a fake window with lighting on one side. Then use large format tiles or cast resin for the floor and walls and the whole toilet instantly has the "wow" factor.

How deep is the room, where is the door, how wide is it and does it open inward or outward?

That looks great – no question. But in our case, tiles and floor are to remain. The only question is whether water can be diverted from the flush cistern supply to a small washbasin and whether the wastewater can somehow be routed back to the toilet drain or directly into the flush cistern again. It is a (cheaply) rented property; the rental period could possibly end again after 2 years and nothing is to be expected from the owner’s side.

That looks great - without a doubt. But in our case, the tiles and floor are to remain. The only question is whether water can be diverted from the cistern inlet to a small washbasin and whether the wastewater can somehow be routed back to the toilet drain or directly into the cistern. It is a (cheaply) rented property; the rental period might possibly end again after 2 years and nothing is to be expected from the owner's side.

In short: Nothing comes from nothing, without interventions it will be nothing. You also have to clarify this with the landlord anyway, otherwise there will be trouble at the handover at the latest. Without interventions, your idea will not be feasible.

There are toilets with mini washbasins in/on the cistern (sometimes even using the handwashing water for flushing). However, I don't know if they are reliable, because I don't think they are installed very often. Ecosia yields a few hits on that. "WC mit integriertem Waschbecken"