1. Is image stabilization a must? It drastically effects the price of the lens.
I'll run counter to many recommendations and say that whether or not you need image stabilization depends on your skill level and the kind of shooting you do. It's most useful at longer focal lengths and to some extent, slower shutter speeds. If you're a wider angle, sunny scene shooter, you'll likely notice less benefit from it. But it's available on so many lenses these days, I don't see much reason to avoid it unless you really, really need to shave cost.
2. What would be your choice for a walking around lens for $300?
If it were me and I was limited to a $300-ish lens, I'd pick the Canon EF 50mm
f/1.4 USM. No, it's not a zoom but it has very good glass for the price point (~$360), it's reasonably fast, and at wider apertures out-of-focus backgrounds melt into a very pleasing "bokeh".
3. Will the average Joe (or below average Jason) notice the difference in picture quality between a basic lens and an USM?
As mentioned, USM doesn't directly affect image quality other than potentially affecting you ability to focus quickly and in some circumstances, more quietly.
I'll also buck another recommendation or two about investing glass. I still shoot my old Canon 20D, which has the same sensor form factor (APS-C) but lower resolution than yours. Most of my shooting is with the aforementioned 50mm and a 24-105mm
f/4 L IS USM "luxury" lens. Next on my wish list is the 70-200mm
f/2.8 L IS USM. The camera is ancient by today's standards but it still benefits from higher quality glass even with the lower resolution.
Speaking of high resolution, more modern sensors have driven the introduction of several L series Mk II lenses because higher sensor resolution has gotten to the point of revealing previously unnoticed optical defects...or so the perception and/or marketing schtick goes. When I get around to buying a new lens, I'll be looking for a deal on a pre-Mk II lens. You might be able to find a deal, too. For instance, my 24-105mm is on Amazon for about $860. I know it seriously busts you budget but IMO, it's a very good walk around lens. If you can (or decide to) swing it, the price only stings for a little while! :whistling: Otherwise, I think you've received some good lens recommendations so far.