Sorta - cats are the principal host for T. gondii as it can only reproduce inside of cats. The altered behavior of mice is one of the ways cats can become infected (by eating the mice with tissue T. gondii cysts). Once infected, cats excrete T. gondii oocysts in their feces that can go on to infect other animals or people. Thats why pregnant people are told to not clean their cat's litterbox due to the risk of coming into contact with oocysts - fetal toxoplasmosis can cause birth defects and spontaneous abortions IIRC.

Cats are often not that great at efficiently killing rodents (a working dog would likely be better for that). None of my childhood cats had much interest in doing more than slowly torturing or dismembering the occasional mouse. I would also personally worry about the spread of toxoplasma from mice to pet cats.

The welfare of cats as dedicated mousers is also often highly questionable at best, depending on the situation. For instance, the most beloved barn cat at my mom's barn was just killed by a group of coyotes as he was essentially allowed to free-roam.

Rodenticides are pretty awful for wild animals. They've also been linked to devastating mange outbreaks in wild cats, such as bobcats, in other states.

No, they are not. All lizards are reptiles, while amphibians are neither reptiles nor lizards. Here is a super simplified vertebrate evolutionary tree. Note that reptiles are more closely related to mammals than they are to amphibians.