As part of the Naturalization Application, you have to show "Good moral character", esp. within the past 5 years of filing. USCIS generally takes the position that the 5-year period starts from the last event of your crim. case, i.e. when ALL conditions are met and satisfied (e.g. probation terminated, etc.). So once you are completely DONE with your case, take that date and start the 5-year clock. Filing sooner is very risky. Consult with a lawyer.
The H-1B petition is filed with USCIS by the U.S. company (petitioner) who wishes to employ the worker. IF and when USCIS approves the petition, the worker can apply for his corresponding visa document (to be issued into the passport) at the U.S. consulate which has authority over the worker's country / place of residence. (Thailand).
Yes, same-sex marriage-based Greencard petitions are now possible under U.S. immigration regulations, provided there is a legal marriage and the petitioner is a qualifying individual.
If your Greencard eligibility was based on marriage to a U.S. citizen, you can only apply for Naturalization after 3 years of the initial grant of Greencard, provided you are still married to that person. If you are no longer married to that individual, then you would be subject to the 5-year waiting period everyone else would be who would not be eligible to apply after 3 years of LPR-status.
Generally, as other colleagues have stated, civil litigation cases in the U.S. (or elsewhere) do not negatively impact one's admissibility into the United States.
Depending on the disposition of your criminal case(s), this could very well have negative impacts on your immigration efforts later, especially at the felony-level. Your visa could be cancelled, you may have problems obtaining other visas in the future (at least for a while), and you will certainly experience issues upon attempted entry into the U.S. (incl. secondary inspections, and possibly likely refusal of admission).
As Jeff Devore stated so well, DO file the I-130 petition(s) NOW and get a priority date (essentially a "date stamp") on those for later. Start that clock now, because on their own, those petitions have to sit there and "kill time". There is no good legal reason (I can think of) to NOT file those I-130 petitions....
While your status in the U.S. terminates on the day that your employment with the U.S. company terminates, you have a reasonable amount of time to get your affairs in order and depart the U.S. If you left within 30 days of your termination, that should be ok. It may help to also have a letter from the company stating that the termination was not due to performance or conduct (if that is in fact the case), to show that the job-loss was not caused by your own actions.
If you can file/amend the 2008 return, go ahead and do that. You may also want to go to the nearest IRS office locally, and order a Transcript of your tax returns. Also, include in your response to the USCIS letter (yellow), any other financial information which might be helpful, such as current W-2/1099, payroll statement, copies of pay checks, copies of recent bank account statement, etc. to show your income/earnings. (also include wife's documentation, if she's working).
The waiver will be easier if you have a U.S. entity with some clout backing your request, e.g. an interested government agency, acting on your behalf. What type of J-1 were you on? Medical? Depending on what your home country is, you might want to contact your nearest Consulate here in the U.S. and ask about their general policy for "no objection" letters. Was the financial aid from a private source or U.S. government source?