Nope. The word "office" isn't part of the official name, presumably.

Try substituting any other thing the Homeowners Association might have. Would you capitalize the Homeowners Association softball team, the Homeowners Association stationery, or the Homeowners Association location?

(I'm a grammar Nazi--but only on request.)

If the title 'Homeowners Association' refers to an actual association, then yes. It is the name of a registered association, and is therefore a proper noun. All proper nouns should be capitalised.However, if it is used as a general term, e.g. "Are you part of any homeowners association?" then it does not require capitals.

It won't. Homeowners insurance is protection from sudden accidental losses, it does cover association dues.

Yes. The homeowners association ID may be valid in any establishment.

If the subdivision is subject to a Declaration of Restrictive Covenants that include a homeowners' association then your property is subject to mandatory membership.