tangrien

Shouldn't have done that.
Really shouldn't have done that. It will just cause confusion at USCIS when they show up, and confusion at USCIS should really be avoided...
If it's actually "express mail" as you say, it may be possible to get both the application package and the checks returned to you: https://www.usps.com/manage/intercept-package.htm
I would suggest doing that before they arrive at USCIS, then resending together.
Or even better, pay for a one-time consultation with an immigration attorney to go over your application package with you before you send it. This ought not to be very expensive, and in your case I think it would be a good idea, because (no offense intended) the face that you made these two obvious mistakes suggests there's a good chance you made some more important mistakes somewhere in the application.

From the USCIS website:
Criteria for Expedited Processing of an ApplicationUSCIS will expedite an application, including an application for a travel document, Form I-131, in certain situations, which may include:
Severe financial loss to company or individual Extreme emergent situation Humanitarian situation Nonprofit status of requesting organization in furtherance of the cultural and social interests of the United States Department of Defense or National Interest Situation (Note: Request must come from official United States Government entity and state that delay will be detrimental to our Government.) USCIS error Compelling interest of USCIS You may maket an expedite request by contacting the National Customer Service Center (NCSC) at 1-800-375-5283 or by submitting a written request and supporting documentation with your application.
Emergency Advance Parole DocumentsIf you are experiencing an extremely urgent situation, you may visit your local office to request an emergency advance parole document. When visiting a local office to request emergency advance parole, you should bring the following items:
A completed and signed Form I-131, Application for Travel Document The correct I-131 filing fee Evidence to support the emergency request (e.g. medical documentation, death certificate) Two passport-style photos.

Unless there is solid evidence to the contrary, it won't be an issue. You will not have to present them with any proof, they need to prove that you did have intent. Even if they do, intent alone is still not enough to deny your AOS. You may be asked about it, but other than that, I wouldn't worry about it.

Definitely attach a note for the years you didn't file taxes explaining why. As long as you currently make 125% above the poverty line for your household size, you'll be fine. Past income is irrelevant.

Yup it was USCIS approved surgeon. Thanks a lot tangrien!

http://www.visajourney.com/content/immigration-parents
This is the link. Congrats on the opportunity of bringing your parents. Please exercise patience on this process