The laws and even the industry is not standardized. Though things are working their way to being more consistent. You should definitely be certified before taking payment for training clients or teaching classes. But to know for sure the regulations in Texas you should contact your state government. In Hawaii I need to register my business and obtain insurance as well as a business license. In Colorado I also needed to these things. I found the right place to contact by searching “starting a business in … Hawaii… Colorado” and got the information I needed very easily. I would caution you not to be fooled by sites that are not official. All the sites I visited in my search were .gov sites. There were several unofficial sites that popped up on the search.
In order to get a job at any fitness location you will need to be certified. As for the insurance part of your question, I believe you do need to be certified if you want the PT liability insurance. If I was the insurance I wouldn’t cover anyone who is not certified. Now, some places might let you train clients and then give you a certain timeframe to get your certification (I don’t agree with this approach, but it is what it is).
this is a question of whether an insurance company will provide you with liability insurance without having a personal trainer certification or equivalent credentials, and the answer is “none that I know of”.
You may need a business license to operate a personal training business http://publicrecords.onlinesearches.com/Texas-Business-Licenses.htm.