Answer "yes" based on being a member of the US Army reserves, and as the computer does not provide a way to add an explanation, I recommend that once you print out the N-400 you write the explanation by hand on the form near the question, and attach a copy of a document to your application showing that you are a member of the US Army reserves.
The answer varies a lot depending on the workload at CIS at any point in time BUT according the CIS's own regulations (which they do not always follow) they must issue a decision within 120 days of your interview (but they do get to add on the amount of time that it took you to respond to the N-14).
I agree with your attorney. It can take more than one year for Immigration to file the NTA with the court. While you are waiting for it to be filed so the Immigration Judge can make the final decision on your I-751, Immigration must keep extending your I-551 stamp.
Yes, you can file for your mom. You will need to include your father in your mom's Affidavit of Support when you are counting your "household" size, so it will make a difference as far as how much income you will need. (If your income does not meet the guidelines, then you will need a co-sponsor.)
By the way, you are legally obligated on the Affidavit of Support either: 1) for 10 years, or 2) until your father becomes a US citizen (which ever happens first). Your obligation does not end when he gets a job.
As a permanent resident, once you file the I-130 visa petition, you might wait approximately 20 months for the priority date on that petition to become current. The website for the current visa bulletin is shown below (look at the Family 2A category). Once the priority date becomes current and you have completed the next step of processing at the National Visa Center, then your wife will be scheduled for an Immigrant Visa interview at the U.S. Embassy. Although priority dates can sometimes move faster, and can sometimes move more slowly, based on the current movement, it seems likely it could take a little more then 2 years to bring your wife to the U.S.
I don't know the answer to whether it would be faster to bring your wife to Canada -- I'll be interested to see what a Canadian immigration lawyer says about that.
Unfortunately, diversity visa lottery winners must use the visa number that was won within the fiscal year that the winning is for. It is a "use it or lose it" situation, so your visa would have been lost when you did not use it in 2012. There is no way to bring it back now.
You can apply for US citizenship if you are married to a US citizen, you have been married to the US citizen for 3 years and have been living with that US citizen (as a married couple) for 3 years. You can do that whether you got your green card through asylum or through marriage.
It is the FINAL conviction that should control whether or not you are subject to the Adam Walsh Act, so the fact that the charges started out as something else should not be relevant. That said, CIS is hyper-vigilant when it comes to the AWA and I cannot guarantee that they will not try to over reach by looking at the charges. If you file an I-130 and I-485 and it triggers a request for evidence related to the Adam Walsh Act, you should definitely get an immigration attorney who is well-versed in criminal law to make the legal argument that the conviction does not fall within the scope of the Act. Better yet, consult with an immigration attorney (familiar with criminal law) before filing the I-130/I-485 so they can review the conviction to see if they agree that for purposes of the AWA it qualifies as non-sexual and not with a minor.
Apply for her now and do not wait until you get your GC. Use the I-730 petition to apply for her, which is available on the government website at www.uscis.gov. It is taking approximately 5 months for CIS to approve those petitions and forward them to the US Embassy, and then another few months for the Embassy to interview her and allow her to enter the US in asylee status. On the other hand, if you wait until you get your GC to petition for her she will probably have to wait an additional one to two years to enter the U.S. after you petition for her. Either way, you may have to explain why you did not list her on your asylum application. Waiting until you get your GC to apply for her does not allow you to skip over an inquiry into why you did not list her on your I-589. Your failure to list her on your I-589 will make CIS suspicious that it is not a real marriage, so I would recommend submitting more than just the bare minimum to show that you are married. Since the situation may get a little more complicated due to the I-589 omission, it might be helpful to have the assistance of a lawyer.
Unfortunately you must wait for your own green card first. If you were married before you were granted asylum (even 1 day) you would be able to immediately petition for your spouse to be granted derivative asylee status (just like your status). But once you have asylum, you cannot petition for a spouse until you are a permanent resident.