Is it possible to over-edit? Or, is there a point where the story is unsalvageable?

Editing is like sharpening a knife. You hone the story like you would a blade, but if you overdo it you’re no longer sharpening the blade but actually weakening it.
When you’re editing, there’s eventually a matter of diminishing returns regarding the time you put into a draft and the quality of the final story. At a certain point, the time that it would take you to read the entire book isn’t worth the handful of changes that you might be making.
For me, as an artist (read—annoying, neurotic, perfectionist), that’s a hard line to draw, and admittedly, I tend to keep going over my work again and again until I’m convinced that it’s the very best I have to offer.
In my view most people don’t need to worry at all about over-editing. They don’t spend nearly enough time on the editing in the first place.

There’s no point at which a story is unsalvageable, but most people aren’t willing to take the time to make the major, or in some cases seismic, changes that would be necessary to tell that story well. It might take less time to start over or write a different story entirely.

Telling a great story always requires five things: the inevitable movement of the story from the origination to the resolution (that is, every event is caused by the one that precedes it), believability, escalation, motivation, and surprise. Apart from grammatical errors and copyediting, these areas are the biggest problems most stories face. So, it’s vital that as you work on your story, you continue to ask:

(1) Inevitability: Are there gaps in narrative logic? Do things happen for no reason—other than that I think they need to in order to make my outline work?
(2) Believability: Is everything that happens believable even if it’s impossible? Does the character act in a way that’s consistent with his or her core attitudes, desires, inner turmoil, and outer circumstances?
(3) Escalation: Are the stakes being raised? Is the danger becoming more imminent or more unstoppable?
(4) Motivation: What drives this character to act? In other words, what does the character want more than anything else?  How far is he or she willing to go to get it? Is every action that the character takes sufficiently motivated by the story events?
(5) Surprise: Do the scenes, the acts, and the story as a whole end in a way that people won’t see coming, but that also follow inevitably from what precedes it?

If you continually ask yourself these questions as you’re working on your story, you’ll find less need to have to start over or to get to the point at which it’s not worth your time to rework the story to salvage it.

Do you use any writing software, and does that make it easier/harder to edit and move things around?

I recommend a program called Scrivener to collect all my ideas and then combine and develop my first draft of the book. After trying a number of other software programs, I’ve found Scrivener to be far superior to them and at a fraction of the price. Check out their site to see the capabilities and uniqueness of the program.
However, as I near the end of the draft, I will often transfer the document to Pages to do the final formatting.
Incidentally, as I work through the various drafts of the book, I print it out in different formats (with different fonts and font sizes) to edit it. I find that this helps me notice things that I tend to overlook when I keep printing out the book with the same format, font, etc.
For me, I find it most helpful to print out the book in single space to edit it early in the process when I’m focusing more on the story’s flow, then 1.5 or double space to do more word-by-word copyediting, and then in single space again at the end of the process to look at the book more through the eyes of how a reader would see it.
Get the tools, nail down the process that works for you, and then go for it. Remember, software is simply one tool at your disposal. Story matters more than anything else, so whatever helps you improve your story is the program you need to use, even if other people think it’s clunky.

Is it possible for an author's writing to stagnate (meaning there's little style difference, thematic difference, maturation, or development of real or perceived flaws in the writing) over time, and what might one do to prevent such a thing from happening?

This is a good question. I think it’s really easy and, in some cases, quite common for writers to, as you put it, stagnate for a number of reasons.

First, most writers will gravitate toward a certain voice, a certain tone and mood that feels natural to them. I know that with my style of writing and editing, I will end up with punchy, fast-paced text that doesn’t include elaborate descriptions. It’s just the way that I’ve developed my voice as a writer. I like to explore different ways of writing—first, second or third person, present and past tense, different genres—but I do have a distinctive voice that will probably come through in all of my stories.

Similarly, the themes of a story will typically reflect what big ideas or questions the author is exploring in his or her life at the moment. I think that will almost always come through in some way in the story. Personally, I read a lot of philosophy which makes me ask different questions about the meaning of life as I work on each book so that there isn’t too much repetition of moral dilemmas and thematic material.

When I write my books I try to have a different flavor and plot for each one. While all of my novels are thrillers, each has a different feel:
The Pawn: psychological suspense
The Rook: techno-thriller
The Knight: gritty crime
The Bishop: political intrigue
The Queen: terrorist activity
Opening Moves: intense suspense
Placebo: science-based conspiracy

By purposely setting out to create a slightly different mood and plot focus, it keeps the stories fresh and helps me avoid writing cookie cutter novels that are basically the same in plot structure.