You have described very well the properties we all want in our gardens; the problem is in the execution of that aim. The Lady Bird Johnson Wildflower Center will recommend only grasses native not only to North America but to the areas in which those plants grow normally. In this serious drought year, the Wildflower Center specialists have been working hard to find a turf that will grow well under the conditions we have.
Addressing some of the ideas you had; Zoysia is a warm-season grass native to China and Japan which can grow in some shade but does not always perform well there. Bermudagrass is native to Africa, not Bermuda, does not grow well in shade, and is considered one of the most invasive weeds in the South. St. Augustine, while not as invasive as bermudagrass, is also native to Africa, and is one of the worst water guzzlers in Texas. It is one of the few grasses that will do well in shade, but since we now know that 40% of our drinkable water goes into watering lawns, we definitely feel that it should not be considered.
Some of your other ideas did involve native grasses, so we want to refer you to some of the information on our website. First, read this article on Habiturf, which has been researched for some time and is now actually on the market. Notice the bar graphs showing how this grass compares for weed resistance and speed of growing with non-natives. Another article is in our How-To Articles Native Lawns: Multi Species covers some of the same materials. Our understanding is that this mix can tolerate 50% shade.
Since we don't know if your yard is full sun or full shade, we can't make a definite recommendation, but after you read our referenced articles, we think that you will have a better idea of what you can do. The main thing we cannot do for you is provide an evergreen grass or groundcover, particularly that will tolerate much foot traffic, even from little feet.