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Put her married name on the AOS application and be certain to include her maiden name in the 'previous aliases' section. USCIS understands that there are difficulties in changing names with limited identification and expired I94's, especially with K1 visas. The name you put on the AOS application will eventually appear on the green card. So make sure you choose wisely, otherwise you'll have to file for a name change on the green card and that will cost you $500 and months of waiting.
Best to do it up front even though you're still attempting to finalize the change.

EAD and AP are approved by the same adjudicator at the same time. They rarely update both statuses. If your I765 is approved, so is your I131. You're good to go.

The issue, provided you've articulated it accurately, is that you did not provide the correct fee to the USCIS. Cashiers Checks, Money Orders and Personal Checks are accepted. DO NOT send cash. The I-130 fee is currently $420 and the I-485 fee is currently $1,070. If you send a Personal Check, ensure it is drawn on a US Bank account. Do NOT attempt to send a Canadian Check with "USD" written next to the amount.
You have TN Status, NOT a TN Visa. It is fine to move from TN Status to I-765; however, it is NOT a renewal of work permission as you CANNOT renew TN Status you apply new each time -- the I-765 is new status / permission to work. And when filed concurrently with an Adjustment of Status, there is no fee for the I-765.

The visa overstay will be forgiven by the USCIS because of a marriage to a USC. The cost will be $420 for the I-130 and $1,070 for the I-485. You should also consider filing an I-765 at the same time to get work authorization prior to receiving their green card. Also the alien spouse should NOT leave the United States; if they do it will result in a ban from the US for several years. They need to remain in the US until they receive their green card. They should follow this guide: http://www.visajourney.com/content/i130guide2

Put her married name on the AOS application and be certain to include her maiden name in the 'previous aliases' section. USCIS understands that there are difficulties in changing names with limited identification and expired I94's, especially with K1 visas. The name you put on the AOS application will eventually appear on the green card. So make sure you choose wisely, otherwise you'll have to file for a name change on the green card and that will cost you $500 and months of waiting.
Best to do it up front even though you're still attempting to finalize the change.

It should also be noted that a short form birth certificate is still valid and accepted by the USCIS. Per Title 8, CFR 103.2(b) "Evidence & Processing" (http://www.uscis.gov/ilink/docView/SLB/HTML/SLB/0-0-0-1/0-0-0-11249/0-0-0-11618/0-0-0-11634.html) specific to "Submitting secondary evidence and affidavits" which is specifically referenced in the I-485; there is no explicit requirement for a long form birth certificate. Although preferred, the short form birth certificate is generally accepted. If you have time and money to request the long form, I'd recommend it. However, it is highly unlikely, if you were born in Canada (not naturalized), that you'll receive an RFE for a short form birth certificate. This is largely due to the fact that the US and Canadian populous databases are highly intertwined and confirming data provided in the I485 form is extremely easy for the USCIS.
That's specific to Ontario. Quebec, as an example, only adds the parents names to the long form document. Virtually no difference between the short and long form documents.