I have a question about the personal morality of lawyers in texas and
grounds for dismissal based on said morality.

Before I was born, my father left my mother when he found out she was
pregnant with me and had nothing to do with her afterwards, thus
forcing my mother to give me up for adoption. When I reunited with my
mother in 2004, she gave me the name of my father and told me he is a
lawyer in Dallas. I did some research and found out where he worked
and attempted to contact him and when he found out who I was, he hung
up on me.

The only was I have been able to communicate with him is through an
intermediary, and even then he has refused to give me any sort of
medical information or speak to me or answer any questions I may have.

Now, seeing as he is a lawyer, is he violating some sort of personal
Code of Conduct in the state of Texas? Seeing as he could possibly
take a case against dead-beat dads as a profession, on a personal
level wouldn't that make him a hypocrite? Should I notify his
employers about this?

Because your father and mother were never married, nor was official
paternity established, requiring your father to contribute to your
support, he is legally off the hook. Yes, it's painful to you now that
he is unwilling to have contact with you, or answer your questions, or
establish a relationship with you, But although his behavior may have
been, or even now be considered morally reprehensible, in reality even
if you did tell people, including his current employer, it would be
unlikely to have much effect beyond embarrassing him. And that, of
course, would hardly be likely to make him any more willing to
reconsider his response to you.

Here's a link to the "Texas Disciplinary Rules of Professional
Conduct," which lay out the ethical obligations and requirements for
Texas lawyers: http://www.txethics.org/reference_rules.asp?view=conduct
 Section VIII, "Maintaining the Integrity of the Profession" is the
part that addresses personal conduct unrelated to the actual practice
of law. In particular, Section 8.04(a)(2)&(3) talk about unethical or
criminal personal conduct. See here:
http://www.txethics.org/reference_rules.asp?view=conduct&num=8.04
However, I seriously doubt whether refusal to acknowledge or have
contact with an adult child will fall under these parameters. I could
be wrong, of course. If you feel you can demonstrate his unfitness to
practice law on the basis of these official rules, then I'd suggest
you contact an attorney for an official opinion, as anything I or
anyone else says here is for informational purposes only.

I'm sorry you've found your father so unresponsive. It must be very
frustrating and sad. Good luck to you.

Best,
Byrd-ga