I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups?
The answer is NO. Any four times punctured sphere admits two involutions, the quotient by which is an orbifold of the signature you describe. Similarly, you can take a punctured torus, and the quotient by the elliptic involution is one of your surfaces. Conversely, you can cover one of them by a torus or a four-times-punctured sphere. Since very few tori are arithmetic, same is true of your class of groups.