Induction cooktop with integrated extractor, experience?

So I only do what really needs water (washing vegetables, for example) at the sink and the rest on the island next to the cooktop (sink and cooktop are opposite each other with about 1.20m in between, so I have to take a step between island and sink anyway). This way the chopped vegetables can go straight into the pan or are distributed on the island into different containers or the like. It always depends on the habits in the processes what and how you want to have things placed.

In the ETW, I have it just as you describe. Even with the same distance. Still, it annoys me a lot that the vegetables drip onto the floor when transferring them from the sink to the stove. That's why I'd rather finish chopping right next to the sink and put them directly into the pan there, or if not everything is going in at the same time, first into small bowls and then over. Of course, you get used to everything, but the question is what is ergonomic. In the new building, I don't have an island, so the problem is completely eliminated, unfortunately, I am then also turned away from the living area all the time. Unless I move the preparation right to the dining table, which again ruins the ergonomics. Well, you have to set priorities; space and the feeling of the living area seem more important to us for now.

and if you "only" make salad? also transfer everything over?

Do you peel potatoes?

There are already sloping ceilings, but the cooktop is in the room.
I found cooktops from AEG for under €1000, so I thought about one of those.

I don't know if going for the cheapest ones is a good idea. They really need to have power; otherwise, you often end up buying twice when you go cheap.
But I can't say anything about AEG myself, you'd have to do some research and ideally find people who have one.
We also talked a lot about this and got good tips. In the end, we decided on a Bora cooktop. But I can't report anything yet; we're still waiting. The Bora units also cost a pretty penny, which you might not want to spend for a rental apartment.

If the cooktop is in the middle of the room, an island hood would also remain an option, right? Those are mounted on the ceiling.

In a rented apartment, I would look for a robust and simple solution; a recirculating hood would come to mind.

But the cooktop on the island is a classic fashion mistake. In general, you need more time on the sink side. So it would make more sense to have the sink + countertop on the island.

Those who cook with and for guests are happy to be able to turn to them thanks to the cooktop on the island. Therefore, for me, it is not fashion, but a way of life.

As I explained above, in an ergonomic kitchen it is not necessarily the case that you spend most of your time working at the hob. If you want to turn to the guests, you would actually need the sink and hob and a lot of workspace on the island. That simply does not always work.