"Have you eaten (or had) your lunch" is past tense and I think would be translated to "Hast du dein Mittagessen gegessen" where "gegessen" is the past tense of "eat." I don't think there is a German equivalent to "Have you had your lunch." They seem to be more specific in their verb choice.
Actually "hast du dein Mittagessen gegessen" is present perfect, and "gegessen" is past partizip (participle). The past tense of "to eat" is "ass" (lol. Well actually with es-tset, but don't have it on my keyboard) while the equivalent of "Have you had your lunch" is "Hast du dein Mittagessen gehabt", but dunno if German-speakers actually use it like that; though grammatically correct, feels a bit odd to me.. :D
Yes. While "Have you your lunch" is a literal translation and sounds like it was used in older English, current English speakers would say "Do you have your lunch" instead. It's similar to how "Ich esse mein Sandwich" can translate to "I eat my sandwich," but current speakers prefer the other acceptable phrase "I am eating my sandwich."
Well actually no. "have" here is the main verb, while "do" is an auxiliary verb. If you want do not want to add do, you can go with "have you had your lunch", but that would be present perfect, instead of present tense.. But since you are asking a question in present tense, you will need an auxiliary verb in most cases..
In the "Possessive pronouns" section it sais that the inflection is the same as for "kein". Go to http://en.wikipedia.org/wiki/German_declension#Indefinite_articles and you'll see that the accusative form of neuter is "kein", so it is "dein".