It is possible but the courts tend to favor what's called joint managing conservatorship. You would need to consult with an attorney to discuss your issue
Getting sole custody requires a court order. You may file a suit affecting parent child relationship without filing for divorce. Call a lawyer and get some help.
I am not intending this to be legal advice, because I don't know the particulars of your situation. Call me if you would like to discuss this or other isues.
It should be. You will need to file for Divorce in Tarrant County, and get your husband served with the papers (that is difficult, but not impossible). Then you need to set a hearing and get a Judge to appoint you as Sole Managing Conservator of your daughter. This won't be as easy as it sounds, especially if there are legal proceedings involving your marriage or daughter in India.
This information is not legal advice and does not form an attorney-client relationship.
Short answer is no. Custody or conservatorship is court ordered. So you would have to file for divorce to do so. Once you file the court presumes that two married people should be joint managing conservators with one party having the right to determine domicile. To become sole managing conservator you'd need to show reasons why it wouldn't be in the kids best interest for your husband to be a joint managing conservator.
Sign up to receive a 5-part series of useful information and advice about child custody law.