The man on "type what you hear" read on the slow speed, ce-ri-ca, three clear syllables, at least to me. Since there are only two vowels I assume there should be only two syllables, cer-ca. However, in Spanish the r's are trilled. Does that trilling separate into a sort of third syllable, ce-rr-ca, or should it be pronounced cerr-ca. A native speaker might be helpful here.
I'm not a native speaker, but I have a good accent and good ear. I don't have the link to that "type what you hear", so I can't hear it for myself. However, "cerca" is indeed only two syllables. I guess there is some "length" to the "r", but it is not a separate syllable.
I went to "words" to listen to it there (normal speed only). It does not sound like "ce-ri-ca" there. Here's that link: https://www.duolingo.com/dictionary/Spanish/cerca-Adverb/f1518c38bc13972c753d8bc85f49e993
The temporary/permanent distinction is an oversimplification, so you should not rely on it too much. The "temporary" mainly applies to things like emotions and physical conditions, like being tired or ill.
"Estar" is used for location, and saying that something is near is a way of saying where it is.
Here's a fun way to remember: "For how you feel or where you are, always use the verb estar!"
When I took Spanish (long ago!), we were taught to put the verb first in questions. But what I see most often in Duolingo is forming a question simply by putting question marks around a statement. In this case, "The city is close" is a statement, and "The city is close?" is the corresponding question.
I thought this way of forming questions was odd, but apparently it's quite common among native Spanish speakers (and I hope that a native speaker responds to this to confirm it).
I did find some information at this link: https://www.thoughtco.com/word-order-in-spanish-sentences-p2-3079445
It says the following: "In Spanish questions, the subject almost always comes after the verb. ¿Escribió Diana esta novela? (Did Diana write this novel?) ¿Qué escribió Diana? (What did Diana write?) Although it is possible in informal speech to phrase a question like a statement as can be done in English — ¿Diana escribió esta novela? Diana wrote this novel? — this is seldom done in writing."
This confirms the verb-subject order as the usual way to form a question, which is what I was taught. It also notes that making a question out of a statement by adding question marks to the statement is something done only in informal speech. That is comparable to English, where a question like "You went to the bar last night?" would be used only in informal speech.