"Did those changes -- from a methane to an oxygen-rich atmosphere for example -- result in radically different RNA or DNA sequences for the bacteria from each epoch?"

It certainly left an imprint. I can't google up a concrete reference, but there are genes in our genome that come from the era when there was a lot more methane in the general environment meant to essentially put the organism in hibernation when they temporarily encounter a certain type of high-methane environment.

How "radically different" it is depends critically on your definition of "radically different". It is certainly detectably different, which is more to my point. Sequencing a Martian lifeform without those markers would stand out and provide very strong evidence that they don't come from a heritage that had that in its past. (I have no idea how widespread those genes are, so I don't know how much Earth life that would eliminate as possible ancestors.)

"Why should different starting conditions result in different forms of basic DNA anyway? Wouldn't the basic problems of evolutionary viability be the same?"

Genes code for chemistry. Different starting chemistry implies different solutions for problems encountered early in life's existence. Not to mention there are multiple solutions to problems, and the odds of life choosing the same initial solutions are just negligible.

You can see this if you play with evolutionary programming, and look at the resulting programs. You can see it in conventional genetic programming, too, if the problem has many unrelated but equally appropriate solutions. In fact, you can see it in that music link we had a couple days ago. Any given evolutionary run tends to create a population that congregates in a relatively small part of the space, regardless of how many other viable paths there are. Evolution's ability to escape local optima is greatly oversold in the popular press.

"So, again: why?"

Because the probability is so long against it. Why would two 100-fair-coin-flip series ever be the same? It's theoretically possible, but you'll never see it.

Basically, what my point boils down to is that a Martian life form won't fit into the phylogenetic tree of life. See http://www.talkorigins.org/faqs/comdesc/phylo.html , which doesn't relate the tree directly to the topic at hand IIRC, but you should see how it relates with some thought. Something that does fit into the phylogenetic tree very probably came from Earth.

"Those that have the combination of both ignorance and intellectual curiosity too often get met with sarcasm and indifference by people who then complain that nobody takes their opinion seriously."

I call foul. You grunted a one word question to a pull quote, a "why" that I answered two paragraphs later. If I had had no answer, I would have understood. This post of yours is not unreasonable, asking for specific further elaboration, but just "why?" didn't cut it. (You prompted me to remember the phylogenetic tree page, which puts my point on much firmer footing, which I appreciate.)

Also, since I can smell the "But you can't prove any of that" coming, I would say, prove for me that life will exactly evolve the same way on Mars that it would on Earth. Actually, I explicitly won't hold you to that, because we're all speculating and handwaving. Even a biochemist of fifty years practice is just speculating and handwaving. Until we start building artificial life forms and start really exploring the space of "valid life forms" neither you nor I nor anybody have the actual data in hand. (We're just barely at the threshold of artificial life and by no means have we even begun exploring the space of possible life.) However, if you go back to my original post, I actually make a judgment call on probabilities, so it's not like that's news to me that it won't be proof.