I had to check what that thing is first - it doesn't seem to be anything sauna related.

Do you only have that thing, or also a proper sauna oven? If you do have an oven I'd just use that for the steam, if not you might want to start by searching data sheets from the manufacturer - I'm not sure if you'll be able to find useful tips for your unit here.

I don't have experience with the company or that sauna - though it looks pretty low on those pictures, and it seems to have two rows of benches roughly at the heights where a finnish sauna would have the first/second row, so the third one is missing.

As for the connection, you can have the cables laid inside the walls - but unless somebody thought about that and prepared a duct that'll mean cutting a deep slit along the complete route, putting the cable in, and closing the wall again - it's dirty, loud and expensive. So unless there's a good reason for putting the cables inside the wall on top is often the more sensible option.

1983 aka 40+ years old

Not really such a big surprise - my sauna has the original Helo heater from when the house was built in 1979. The timer recently started to get a bit opinionated - but other than that it's working fine.

Do you have a picture of the controls? No type sign or anything with some more specifications on one of the other sides?

Do you have details about the exact material? "Plastic" doesn't really say much - plus you don't really have to worry about the melting point, but the glass transition temperature. That's the temperature at which it loses stability, which can be more than 100°C away from its melting point. Depending on the type of plastic and additives you're looking at a range of roughly 50°C to 400°C for the glass transition temperature - and I'm assuming that kind of part needs to have a relatively high rating for fire safety reasons. So, try to get the datasheet for that thing.