"Isn't the current theory that all life on Earth has a common ancestor? Wouldn't this point put that into question"

Not really. For one thing there's too much commonality across life in terms of things like the genetic code, i.e. what a sequence of genetic letters "means". The genetic code is arbitrary but essentially universal with minor variants.

For another thing, the organisms now present can be separated from each other by as much as 8 billion years of evolution (4 billion on each branch) meaning there's no particular reason to expect some shared gene to be present everywhere. New genes arise as mutations from old ones, old genes become obsolete and get removed, etc.

It's vaguely like how, if you fork a coding project, and the two projects are allowed to continue indefinitely, eventually there might be no code in common, even without a complete rewrite ever happening.

Deep down, somewhere in the genetic code, it still says "Mozilla (compatible)"

Not when your browser fork is re-purposed for something else almost entirely, like an inline image or documentation viewer. As I understand it, organisms often have novel uses for old tools.

That explains why a human being is like a gecko.

I would guess that the earliest life didn't accidentally hit upon optimal or even reasonably good ways to do whatever is needed to reproduce.

If that's the case, it isn't that unlikely that every evolutionary branch managed to improve (or discard) every part of the original machinery in some way.

Depends on what kind of life you consider. It's true for mammals. It's true for animals. It's probably even true further up our ancestry. But apparently it doesn't mean that "all things that have genes" share a subset of genes from the same common ancestor. As I understand it that's because they share a common ancestor that didn't have genes (but at that point I'm not sure whether "ancestor" is even the right word).

Only if all life converges on having dna. If there are multiple origins of life that survive there would probably be multiple different self replicating molecules.