"Where is the closest convenience store?"
いちばんちかい is acting as a modifying clause to コンビニ in this sentence. Modifying clauses, sometimes called descriptive subordinate clauses, modify the noun that follows them. Think of them like adjectives, except with full phrases (sometimes including a verb) and not just a single word.
Modifying clauses are also one of the few grammatical points in Japanese where the word order isn't particularly flexible -- the clause always precedes the noun it's modifying. Hence why your sentence was not accepted.
The way you phrased it here would be understandable, probably, but it's very awkward and not technically correct.
Yes, unless you're writing only in kana. Mottomo is more formal being an exception to the rule: https://japanese.stackexchange.com/questions/38627/difference-between-%E3%82%82%E3%81%A3%E3%81%A8%E3%82%82-and-%E4%B8%80%E7%95%AA . . .