Qualifying for a public defender is based on income and assets. Have you tried to consult with a local criminal defense attorney, or are you assuming the retainer would be so high? If you haven't done so, set up an initial consultation with a local criminal criminal defense attorney.
By PTD, do you mean Pretrial Diversion and the case was dropped? Or did you plea nolo contendre (no contest) and get probation and a withhold of adjudication? Either way, the answer to your question is no.
A background check will usually catch a misdemeanor. An expungement would be beneficial. Contact a local criminal defense attorney for an initial consultation. Only some records are eligible foe sealing and/or expungement.
If this is your first court date since being arrested you will probably get a continuance if you ask. You should schedule a consultation with a local attorney as soon as possible. It's always better to go into court with an attorney.
The child support may be modified even if the judge does not grant the modification of time sharing. If your income went up $25K, it sounds like you have a substantial change in circumstances which may result in your child support being raised. I must respectfully disagree with my colleague regarding the standard for child support being need and ability to pay. That is the standard for spousal support. Child support is based on statutory guidelines that takes into account both of your incomes.
Petitions for parental relocation are the most contested issue in the family law realm. It sounds like you have some good arguments. Consult a local experienced family law attorney to discuss this case further.
I agree with the above answer. Sometimes an arrest can happen several months after the accident while the police wait for blood and/or urine test results. Consult with a local experienced criminal defense attorney ASAP. Do not discuss the facts of the case in a public forum such as this.