I do have good news.
And it's about damn time. Today, after yet another conversation with a tech support agent who did in fact verify my warranty, send the message to the repair center, and got them to start fixing my computer. Thank you, Ray. You are by far the most superior rep Sony employs (even moreso than Glendal).
When I logged onto Sony's website to check my repair status, I finally got a message that didn't tell me to pay $726.73 for the repair.
The Sony product that you have inquired about is currently with a technician. The repair has not yet been completed. Repairs take approximately 10 working days to complete.
Hey, at least they are working on it now. Before I couldn't even get them to acknowledge my warranty (which, as it turns out, DOES exist). Who knows when I'll actually get the computer back from the repair center, and who knows if anything saved on it will have been preserved. All I know is this mess is one step closer to over.