My house is in a "postal district". It's also in an "electoral district". But outside of those narrow technical terms, I don't live in a "district", I live in an "area" or "neighbourhood" - we have good public transport in my area, there's a really good sushi place in my neighbourhood, the youth-club in my area runs a popular soccer camp during the school holidays.
While I would use ceantar to translate "district" into Irish in most cases, "district" isn't an approriate English translation for every example of Ceantar - a "nuclear-free zone" is ceantar neamhnúicléach, for example. And the Irish for the District Court is an Chúirt Dúiche (Ireland is divided into 23 districts for judicial purposes - they are not the same as the 39 Dáilcheantar or Dáil constituencies).
Personally, no. Admittedly I hadn't realised that the rules surrounding the use of a/an before h/u are based on pronunciation rather than any strict rule. I had previously thought one would write 'a honest' but say 'an honest'. I'm learning as much English here as Irish!
Similar to the difference between 'a uniform', but 'an umbrella'. Depends on whether the letter is pronounced as a vowel or not! Applies to Y as well. Plenty of consonatal Y words, but the only examples of words beginning with a vowel Y that I can think of are 'an Yttrium atom', or 'an ymbryne'.
i mo cheantar is quite specifically "in my area/district", and while things that are "in your area" are "near you", i mo ceantar doesn't mean "near me".
in aice le is one way to say "near", but tá ostán in aice liom suggests to me that you are standing right next to the hotel, not 5 minutes walk from it, but I would use Tá óstán in aice láimhe to say "There is a hotel nearby".