You will be able to adjust after the refiling if you are in legal status. Filing an appeal or motion to reopen DOES NOT give you lawful status.
If your employer retained an attorney for this, speak with that attorney.
No attorney, whether practicing immigration law, personal injury law, or criminal law, will EVER counsel someone to lie on any application, whether it is to the US government or a foreign government. It should therefore be evident to you that no attorney, whether on AVVO, some other website, or in person, will even suggest you should lie.
Aside from that, it is highly unlikely that you would be able to cover your tracks as well as you think, and once your lies are revealed, you wll have more...
There is no fixed time period for the USCIS to adjudicate your application, even if expedited. I would suggest that you wait no less than 90 days from the date the expedite request was granted to inquire further.
As you are not the petitioner, you have nothing to do with these petitions and cannot withdraw them. Only your wife has the power to do that. As to the income requirements, that will not come into play until several years from now, when further documentation will be requested by the National Visa Center.
As I previously answered, it is possible that you will receive an RFE (Request for Evidence), but not definite. It will depend on several factors, including the adjudicating officer's belief that it is needed. It is not a good idea to send in the copy unless you receive an RFE; it is too easy for it not to get matched with your application. Wait to see if an RFE is issued.
Whether you can travel safely depends on your conviction record. You could face problems depending on the charge for which you were convicted. I would STRONGLY urge you to consult with an experienced immigration lawyer near you as soon as possible BEFORE making any trip out of the country. Why risk problems if they can be avoided?