TASK DEFINITION: Given the Question and Answer pair, detect whether the answer is acceptable or not. Denote acceptable answer as "yes", otherwise denote as "no". An acceptable answer is consistent with the context in question, detailed, long, and uses general knowledge of English. Answers that are not definite and suggest a variety of options are not acceptable answers.
PROBLEM: Question: If you, for example, have to add a person to an application whilst that person is on the phone, how do you politely ask for that person's gender if the voice and/or name has not proven decisive? To clarify why I posted this question on this site: I was looking for an answer within the context of the English language and usage. The given answers have proven that this question really isn't off-topic, other languages know different distinctions that are relevant to addressing this issue. 
Answer: I totally agree that this is not really about English language usage but here's a suggestion anyway...preface the question with an apology, "I'm sorry, but I'm required to ask this question, even if it seems obvious, may I have your gender please?".

SOLUTION: yes

PROBLEM: Question: The word year when pronounced starts with a phonetic sound of e which is a vowel sound making it eligible for being preceded by an. Yet, we tend to write a year. Why? 
Answer: I reject your premise that the word year starts with a phonetic sound of e. Here's how it is pronounced according to the various dictionaries: Wiktionary: (RP) IPA: /ji/, /j/, SAMPA: /ji@/, /j3:/; (US) enPR: yir, IPA: /ji/, SAMPA: /jir/ Merriam-Webster: \yirAmerican Heritage Dictionary of the English Language: /yir/ Collins English Dictionary: /j/ Words that start with the /j/ sound are preceded by an a, not by an an. Compare: a user, a utility, a yak.

SOLUTION: yes

PROBLEM: Question: I recently had somebody tell me that a mutual friend of ours who is genderqueer prefers that people refer to him/her using the gender-indefinite pronoun they. In some cases, this almost seems okay: Kris left their umbrella at our house. On the other hand, if Kris is sitting right next to you, it feels very odd to say They (meaning just Kris) would like more cake. Or even odder, Kris would like some more cake, can you please pass it to they/them?" Are these usages grammatically correct? Are they in the process of becoming grammatically correct? Are there more correct alternatives? 
Answer: I think the reason for your friend's preference is that using either the male or female pronouns implicitly pigeon-holes the person in question as either one or the other. However, all of the examples you give seem to me to be forced, and to shout out loud "Hey, look at how sensitive I'm being! I'm not calling Kris either male or female!" There are sensible alternatives to all of these examples that do not break any grammatical rules. "Kris left their umbrella at our house" : Kris left an umbrella at our house (yes, it could technically then be somebody else's umbrella that was left behind, but would you really know?) "They [Kris] would like more cake." : Kris would like some more cake. "Kris would like some more cake, can you please pass it to they?" : Could you please pass the cake? Kris would like some more. While my dictionary (Chambers 1990) does have a secondary definition of "they" as "he or she", it also says that this usage is "with pl. verb", as in "there are lots of people; they are happy". "They is" would not be correct.

SOLUTION:
yes