If it's a CMC with a case manager it's usually pretty easy. Call the case manager and let him or her know you would like to appear by phone, then file a motion for telephonic appearance. I would also suggest consulting with a local family law attorney.
Legal expenses are usually awarded when one spouse is in a much better financial position than the other, or when one spouse unnecessarily causes excessive legal fees. If one spouse wants a divorce, you will be divorced. I do not know of any statutory or case law that requires a spouse to pay legal fees just because the other spouse did not want the divorce.
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 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.