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 you were never married to the mother, you should file a petition to establish paternity. The court will establish a parenting schedule and child support. Consult a local experienced family law attorney.
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.
The party may file a Motion for Temporary Relief to get the child support lowered. It seems to me that would be like trying the case twice. You should consult with a local experienced family law attorney to discuss your case. Many provide free initial consultations.
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.