Re: dogs cannot become cats
> Yet if evolution is true, then it must be possible for the descendants of my cat (or my dog) to, over time, become something that is identical to the other.
It is certainly true that, over a long time (millions of years), descendants of one particular type of animal can become something quite different. It is unlikely, however, that descendant lines of two dissimilar species become 'identical', and no theory about evolution ever claims that they would. That, however, rests on what you imagine 'identical' to be. Some descendants of a dog may eventually become 'cat like' in appearance and behaviour. Some descendants of a cat may become 'dog like' given enough time and an environment where that would be an advantage. To be 'identical' would imply the ability to interbreed and have viable offspring that could continue the line. That is not a factor of evolution, but one of genetics.
In fact cats and dogs have been shown to be able to evolve into different appearance and behaviour within recent history. It may have been by 'unnatural selection', ie breeding by humans, but this is not much different in process than would happen in the wild, it is just that the outcomes are different. Originally there were wolves and wildcats, now through evolution (directed by human selection rather than 'natural') we have dog and cats that are domesticated and tailored to our needs or wants.
A few tens of millions of years ago our mammal ancestors were small shrew like creatures, as is observable in the fossil record. Over those intervening aeons evolution has changed those into the forms that mammals now encompass.
You want to ignore the evidence and evolution of thought and claim 'that never happened'. Fine, but that is the definition of ignorance.