Here is the eligibility criteria:
Sec. 411.172. ELIGIBILITY. (a) A person is eligible for a license to carry a handgun if the person:
(1) is a legal resident of this state for the six-month period preceding the date of application under this subchapter or is otherwise eligible for a license under Section 411.173(a);
(2) is at least 21 years of age;
(3) has not been convicted of a felony;
(4) is not charged with the commission of a Class A or Class B misdemeanor or equivalent offense, or of an offense under Section 42.01, Penal Code, or equivalent offense,
or of a felony under an information or indictment;
(5) is not a fugitive from justice for a felony or a Class A or Class B misdemeanor or equivalent offense;
(6) is not a chemically dependent person;
(7) is not incapable of exercising sound judgment with respect to the proper use and storage of a handgun;
(8) has not, in the five years preceding the date of application, been convicted of a Class A or Class B misdemeanor or equivalent offense or of an offense under Section 42.01, Penal Code, or equivalent offense;
(9) is fully qualified under applicable federal and state law to purchase a handgun;
(10) has not been finally determined to be delinquent in making a child support payment administered or collected by the attorney general;
(11) has not been finally determined to be delinquent in the payment of a tax or other money collected by the comptroller, the tax collector of a political subdivision of the state, or any agency or subdivision of the state;
(12) is not currently restricted under a court protective order or subject to a restraining order affecting the spousal relationship, other than a restraining order solely affecting property interests;
(13) has not, in the 10 years preceding the date of application, been adjudicated as having engaged in delinquent conduct violating a penal law of the grade of felony; and
(14) has not made any material misrepresentation, or failed to disclose any material fact, in an application submitted pursuant to Section 411.174.http://www.statutes.legis.state.tx.us/Docs/GV/htm/GV.411.htm#411.1711
Here is a link to Penal Code sec. 42.01, Disorderly Conduct, which does include Class C misdemeanors:
Sec. 42.01. DISORDERLY CONDUCT. (a) A person commits an offense if he intentionally or knowingly:
(1) uses abusive, indecent, profane, or vulgar language in a public place, and the language by its very utterance tends to incite an immediate breach of the peace;
(2) makes an offensive gesture or display in a public place, and the gesture or display tends to incite an immediate breach of the peace;
(3) creates, by chemical means, a noxious and unreasonable odor in a public place;
(4) abuses or threatens a person in a public place in an obviously offensive manner;
(5) makes unreasonable noise in a public place other than a sport shooting range, as defined by Section 250.001, Local Government Code, or in or near a private residence that he has no right to occupy;
(6) fights with another in a public place;
(7) discharges a firearm in a public place other than a public road or a sport shooting range, as defined by Section 250.001, Local Government Code;
(8) displays a firearm or other deadly weapon in a public place in a manner calculated to alarm;
(9) discharges a firearm on or across a public road;
(10) exposes his anus or genitals in a public place and is reckless about whether another may be present who will be offended or alarmed by his act; or
(11) for a lewd or unlawful purpose:
(A) enters on the property of another and looks into a dwelling on the property through any window or other opening in the dwelling;
(B) while on the premises of a hotel or comparable establishment, looks into a guest room not the person's own through a window or other opening in the room; or
(C) while on the premises of a public place, looks into an area such as a restroom or shower stall or changing or dressing room that is designed to provide privacy to a person using the area.
(a-1) For purposes of Subsection (a), the term "public place" includes a public school campus or the school grounds on which a public school is located.
(b) It is a defense to prosecution under Subsection (a)(4) that the actor had significant provocation for his abusive or threatening conduct.
(c) For purposes of this section:
(1) an act is deemed to occur in a public place or near a private residence if it produces its offensive or proscribed consequences in the public place or near a private residence; and
(2) a noise is presumed to be unreasonable if the noise exceeds a decibel level of 85 after the person making the noise receives notice from a magistrate or peace officer that the noise is a public nuisance.(d) An offense under this section is a Class C misdemeanor unless committed under Subsection (a)(7) or (a)(8), in which event it is a Class B misdemeanor.
(e) It is a defense to prosecution for an offense under Subsection (a)(7) or (9) that the person who discharged the firearm had a reasonable fear of bodily injury to the person or to another by a dangerous wild animal as defined by Section 822.101, Health and Safety Code.
(f) Subsections (a)(1), (2), (3), (5), and (6) do not apply to a person who, at the time the person engaged in conduct prohibited under the applicable subdivision, was a student younger than 12 years of age, and the prohibited conduct occurred at a public school campus during regular school hours.
(g) Noise arising from space flight activities, as defined by Section 100A.001, Civil Practice and Remedies Code, if lawfully conducted, does not constitute "unreasonable noise" for purposes of this section.http://www.statutes.legis.state.tx.us/Docs/PE/htm/PE.42.htm#42.01