You have every right to ask such questions. Here are some suggestions:
What is your training - What led to your becoming an elder? Had you read the complete Bible by yourself before becomiing an elder. How many times have you read the Bible cover to cover?
What training have you had by the WTBTS (note: besides baptism, there is no time that someone has to answer questions as to what they believe) and so any "training" is listening to and giving talks. He has never had to take any type of exam or has never really had to proven himself as to his qualifications. Most likely he commented at meetings, gave talks, went out in field service and met the national average (but may have never actually conducted a Bible study with anyone, and who really knows if the time he reported was spent preaching or eating out) and was seen as others as fitting the idea of being exemplary- He did not have a beard, or questionable length of sideburns, did not have a haircut that may confuse others as to him being worldly, no one had seen him watching R rated movies, etc.
When he tells you elders are 'spirit appointed' ask him what that means EXACTlLY, and then read what Ray Franz has to say about the matter in Crisis of Conscience (I can give you the page number once you have the book).
Has he ever studied the Bible or religion from any perspective outside of JWs. Just having been in another religion does not count.
Does he have higher than a high school/vocational school education?
As far as women are concerned, most likely he will refer you to publications. But men and women are expected to listen to the "counsel" and "direction" of the elders, no questions asked - because they are spirit appointed.
When it comes to personal problems that occur - what does he do (not what are elders supposed to do, but what does he do, and ask for references - people that have actually been encouraged and built up by him using the Bible, not just being nice enough to take time out of his busy schedule to visit them). Elders will always direct people with problems to study the Bible more, attend all meetings so they can be built up, and lean on Jehovah in prayer - ask him what happens when people are doing these things but still having problems.
Since he is supposed to be a "counselor" of God's Word - ask him what counseling skills he has and how he obtained them.
Ask him and each elder you come into contact with.
There is no excuse for them not to answer any of these questions.
