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.