There are actually a number of groups of asexual lizards like these. In the U.S. and Mexico, we have the genus Aspidoscelis (originally Cnemidophorus), known as whiptail lizards. There are about a dozen asexual species, each representing the hybridization of of a parrticular combination of sexual species. Some of the asexual species are even triploid, having chromosomes from three different species. (Most animals are diploid, with one set of chromosomes from each parent.) In Europe, they have the lacertid lizards. Interestingly, the U.S., European, and now these Vietnamese species all look quite similar - don't know what that means.
In answer to some of the ideas you bring up, for the U.S. species most the asexual lines are probably thousands rather than millions of years old. Some species appear to consist of just one lineage (that is, all living individuals arose from a single original hybridization), while others appear to have had multiple hybridization events producing a variety of clones. It appears likely that, on average, members of asexual species are not quite as fit as members of sexual species - but asexual species have an advantage in reproductive rates. When everyone's a female, that means everyone lays eggs. In a sexual species, half the population just knocks up the other half but doesn't actually make any young. So, all else equal, all female populations reproduce twice as fast as sexual ones.