I agree that the simplest thing is to just call each other by name. If you refer to each other as husband and wife in a legal document, that is one step toward becoming common-law married. I am assuming you do not want to be legally married again, since you haven't done so, so do not refer to each other as such in your wills. You can put as much detail as you want to in the wills, including stating that, since your previous divorce, you reside together but are not legally married.
Most couples end up with debt, at least a mortgage, when going through a divorce. Typically, the debts will be assigned to a party in the final decree, but you will need to make sure to have the real estate documents prepared and signed addressing who keeps title to the house, what happens when it is sold, and who is responsible on the mortgage. Remember that the mortgage company will generally not simply release one party from the note without the home being sold or refinanced.
I agree that the Court must have considered some other income or didn't look at your expenses as deductions. You shouldn't be paying such high child support if you really only net $2000 per month after mandatory expenses. If you are going to pursue an appeal or new trial, there are very strict timelines; you need to meet with a local attorney to discuss this immediately. A court will not modify your child support in future without proof of changes to your income, just based on the original...
If she has court-ordered visitation, you will be violating the court order yourself if you refuse her visits. If she is frequently missing her visits, try going back to Court and getting an order that cancels her visits if she misses several in a row, or states that she does not get her visit unless she confirms with you in writing in advance.
The answer to this really depends on whether the mother got court orders giving custody to her parents, or whether that was an informal agreement. If there are no court orders, you just file a Petition to Establish Parentage and ask for genetic testing. If you find you are, in fact, the father, then you proceed accordingly to establish visitation and support.
If a court order was entered, you're going to need to address whether service was correctly attempted and other procedural issues, so...
Your attorney is probably legally correct, but if the insurance company will not release the funds, that is not much help to you. Since you have an attorney already, talk to him or her about what to do next. In these situations, a letter from your attorney's office clarifying how Texas law works often does the job. If not, you'll need to decide if it's worth opening an administration just to collect the funds.
If there is an order for mediation signed by the Court, it should be in the registry. If mediation was successful, the signed mediation settlement agreement (MSA) should also be entered in the Court file (although that sometimes takes a few days). A verbal order to mediate or informal settlement notes will almost certainly not appear in the court file.
An attorney does not need to be specialized to handle a probate case. However, I would strongly suggest you retain one that has practiced in Probate Court before and knows something about the process. There are specific rules and procedures for probate, and it could end up costing more and taking longer if your attorney handles things the wrong way to start with.
Child support will stop when there is a court order terminating the child support. If you are in agreement with it, you could enter that order at any time. Child support doesn't have to stop with termination, but in practice it always does.