You have not listed anything coming close to being "illegal". Probably a good idea for you to consult with counselors for the kids.

How is it possible to have a non custodial parents rights terminated after three and a half years of no contact with my child?

Yes- this is possible. Not guaranteed to work and not a do-it-yourself proposition. You will need an attorney to handle the termination and the...

You can put them on notice to repair then sue to be released from the lease contract.

Sounds like you know this is not his baby. Get legal advice and get this handled correctly.

Can I move out at 17 in Texas if I have a stable home to stay at, a job, and have graduated highschool?

You don't emancipate until you turn 18. If you leave and he calls law enforcement then LE may or may not try to make you return home. You might...

Is it legal for the non-custodial parent to have someone UN-related to our child watch her, on his appointed weekend.

Unless the decree specifically forbids this it is legal. Your remedy is to file a lawsuit seeking a modification of the decree. I advise you to...

Where do I go from here? What am I supposed to do to have his rights terminated when I do not know where he is?

In three years and some months the child will emancipate and this will be a moot point. Why is it so important now if dad is completely absent?

Can I change my last name? Is this marriage valid?

Seems that you would encounter a problem in meeting all of the requirements of an informal marriage since you are not living together.

Can one spouse set up separate bank account during divorce proceedings and keep the funds out of mediation?

This should have been addressed at mediation. If this account was hidden from you then your attorney should address that issue.

Can I sue H&R Block?

If it's a tax that you would have owed, and they are covering the penalty and interest then you don't really have a case.