I am trying to get the logic: is an armchair a deep object in Hungarian, so that one does not sit on (onto) it, but in (into) it?
Yes, something like that.
There are two main types of furniture that I can think of that might be an armchair to an English native. Correct me if I am wrong.
A (close to) normal chair that has arms. For example, an office chair with armrests. - That is a "karosszék" ("arm-ed-chair").
A single-person piece of furniture, a "one-person-sofa", in your living room. Cushiony, roomy, very comfortable. - That is a "fotel".
You sit in(to) a "fotel", and you sit on(to) a "szék". And you can sit in or on a "karosszék".
Ignoring the subtleties of Hungarian (which is obviously much less vague about "where" actions take place) I find it interesting to see the variability in how English is taught and spoken around the world- in Australia it is perfectly normal to sit "in" a chair whether it is an armchair or not- does it sound unnatural to North Americans or others?
In America, as a general rule, if the chair has arms or otherwise encloses the person sitting thereon (armchair, a chair at a table, etc.), you say "in". Otherwise, you can use either "in" or "on" (e.g. a chair out in the open). Except for sofas, toilets, and stools - those are always "on", as are pretty much anything you can sit on that aren't chairs and are in fact not necessarily designed for sitting on, like stairs, railings, cars, porches, tables, logs, etc.
When it comes down to it, there actually isn't a concisely explained rule, or even any rule really. But at least we're consistent! :P
I think that description covers it nicely- it's exactly the same in Australian English.
How do you pronounce karosszék? Is it like karos-szék, karos-sék, or karosz-szék?
Karosz-szék. It's an English 's' sound, but you hold it longer, as with any other double letter. When a letter that is comprised of two letters (a digraph), such as sz, gy, etc, is doubled, only the first part of the digraph is doubled in writing.
Adding to the above, we have to be clear on this. Here we have two different letters that, together, look like a doubled letter. This word is made of "karos", meaning "with arms", and "szék", meaning "chair". So, this is not a double "sz", but rather an "s" and an "sz" next to each other.
So, you can actually aim to pronounce the separate "s" and "sz" sounds, but don't push it because they should actually be mushed together a little bit.
There is a similar looking word, "karosszéria" (a car's chassis, body), which you do pronounce as a double "sz" sound.
Sorry, but there are a few of this "false double letters" that are just two unlucky letters ending up being neighbors. You may have already met "egészség". Well, what two letters may be present in there? :)