Surely "I am staying at home" should also be acceptable, as Hungarian does not distinguish between the simple and the continuous and the latter would be more normal in English.
Which would be fine if it were part of a conversation and the context was clear. However, it isn't.
It's the nature of the verb to refer to the future even if you just say it in itself, "maradok" (I'll stay). You can use the past tense (Otthon maradtam) to mean you have stayed at home, meaning you are still at home. It's not like any other verb like "ülök" that means "I sit/I'm sitting" and if you want to, you can use it in the present to refer to the future like "Leülök" (I'll sit down) The only way to make it mean the simple present is to add words to it that refer to general habits. (Minden nap otthon maradok) (Otthon szoktam maradni.) So without any context, it refers to the future. (I know you can use "I'm staying" as a decision too like "- Are you coming? - No, I'm staying" but that's a different story.)
Not entirely true. When I say "Én itt maradok", that is not referring to the future, but now. Also, if it refers to the future the context will help or there will be something in the sentence that supports this interpretation "Én itt maradok addig, ameddig...". We use the present simple and continuous to refer to the future in exactly the same way in English and it is the context that tells us how to interpret it. It is unhelpful and ambiguous to have a sentence that could mean both and assume one over the other. Your use of prefixes on some verbs is a completely different grammatical point that can be translated many ways in English, it spans both the continuous and perfect aspects of a verb, but many other meanings too. So, for "ülök" it is simply "I sit", whilst "leülök" would be "I am sitting down". The meaning changes from a static event (like a picture) into an active movement (like a film). But it isn't even that simple. Anyway, without context, you simply can't assume any meaning for a static sentence.
Just for the record if a learner happens upon this. "Épp" doesn't make it continuous, it's the word order. Épp ültem le and épp leültem don't mean the same. "Épp ültem le" = "I was sitting down." In "Épp leültem", "épp" means that the action has just finished. "I've/I had just sat down." You can use "pont" for that too. The reason I said "épp ültem le" and not just "ültem le" is because we prefer to say it that way because it sounds nicer than just "ültem le".
"Én itt maradok" is the future. The future is future even if it's just one second away. It's not like "eszek valamit" that can both mean "I'm eating something" and "I'll eat something" as a decision for the very near future (1 second away). You can't use "itt maradok" to mean "I'm staying here" as in "I'm here" if you see my point. My point is that the verb "maradni" works a little differently than any other verb. By the way "I'm sitting down" would be "ülök le". "Épp ültem le amikor kinyitotta az ajtót." = "I was sitting down when s/he opened the door." "Leültem amikor kinyitotta az ajtót." = "I sat down when s/he opened the door". "- Fáradt vagy? (Are you tired?) - Igen, leülök. (Yes, I'll sit down)" It can't mean "I'm sitting down". It's future for the very near future.)
Obviously you meant "eszem" and not "eszek" as this is an "-ik" verb. I know people say it, but you are meant to be helping people speak correctly. The colloquial stuff they can learn later. Your explanation for "leülök" and "ülök le" is completely spurious and incorrect. The reason the prefix moves in the latter was due to "épp" having the emphasis. The sentence you gave is another problem entirely. You are just jumping around from one grammatical problem to the next without explaining or demonstrating the basic ideas in expressing time through verb conjugations. Leülök és megvárom a poént
As I was saying, the verb "maradni" acts a little differently than other verbs. It tends to refer to the future by default and can't really be used to mean the present. It can be used for the second question as a general habit but then you either need the question for context or add other words to it like "Minden nap otthon maradok". When it comes to the first question, you have to say something like "Itthon tartózkodom" (little too formal) or "Itthon vagyok" or use it in the past tense "Itthon maradtam" to mean that you HAVE stayed at home so you're still at home. "Otthon fogok maradni" has a strong emphasis on "fogok" and feels like it's further away in the future.
I would also like to add that I was replying to questions phrased in English. "hol maradsz?" also refers to the future, it can't be used to ask about your current whereabouts. If it's about staying at a hotel or such place, it would probably be expressed as "hol szállsz meg?" The Hunglish corpus (a collection of translations from literature and subtitles) is a good place for learners to check things like this. It' also commonly used to mean "fall behind/late and not where one should be". Check translations here:
(EDIT: Direct linking doesn't seem to work for searches so just search for "where are you staying?" in English then "hol maradsz" in Hungarian on hunglish.hu. Use the quotation marks for the closest possible hits.