Are There Any Situations in Which a Civil Litigant Has a Right to Appointed Counsel in a Texas State Court?

We're not talking here about criminal cases. We're not talking here about federal cases. Though Texas juvenile proceedings are designated as civil instead of criminal, the criminal counsel appointment rules apply.  Sexually violent predator civil commitment is supposed to be civil-- it may oversimplify to say that the criminal counsel appointment rules apply (a criminal defendant has to be competent to stand trial, an SVPCC respondent does not, and such a person can have a guardian ad litem)  and in Texas civil proceedings, except for a very few statutory exceptions, the default rule is that civil litigants do not have a right to appointed counsel. 

 In one in a million cases, a judge might appoint counsel for an unrepresented party-- there is no budget for this kind of thing, but a judge might ask a lawyer buddy to take a case on a pro bono publico basis. 

Let's start with regular old civil lawsuits, money or property or injunction suits.

Now, though, appointments might come for minors or adult incompetents and have to be paid for by one, some, or all of the parties. 

But indigent parents in state-initiated proceedings to terminate a parent's rights have the right to counsel, and courts have the duty to inform the indigent parents of this right.

Plastic Disks No Longer Routine for Appellate Records

Now attorneys-in-charge can get access to documents in their appeals straight from the website of the relevant appeals court, including clerk's records and court reporter's records. Now appellate counsel won't have to get the record from a trial court clerk or an appeals court clerk. Many hours of time will be saved. Some people have complained that one has to sign up one appeals court at a time, but there is a way using a control key to sign up for all the ones you need. The Fourth Court of Appeals (San Antonio) and the Thirteenth Court of Appeals (Corpus Christi and Edinburg).have not adopted the portal.
As files became electronic, it seemed crazy to have to get a pony disk from a live person, often far out-of-town,
You know how your parents tell you how difficult it was to get to school when they were kids. Now, when you are a few decades older you will be able to tell baby lawyers that you actually had to arrange for plastic disks to be gotten to you, and then returned, and they will vacantly gaze at you as a true relic of the dark ages.
(When I was a law clerk, the word processor they had me use was bigger than a kitchen stove, and to start, the screen would roll up the CRT to look like a sheet of paper, so as not to frighten the legal secretaries who were the initial users of these monsters. Personal computers weren't used in the office, and email was extremely rare. There was no Westlaw or LexisNexis or anything like them in the office. Worker's comp cases could be tried. Treble damages and attorneys' fees were mandatory in deceptive trade practice act cases. And practically all the statutory hedges against plaintiffs' getting their actual damages hadn't been built yet. The Health Care Liability Act was just getting started.)