I always think it's a tragic waste when someone buys a DSLR and then wants an all-purpose lens. I'll explain.
It's not possible to make one single lens that does a good job at everything. It's relatively easy (well... easier) to make a lens designed for one task. For example, if I need to make a 50mm lens, then I can put the various optical elements in a configuration and spacing that works extremely well for the 50mm focal length. If I want my lens to "zoom" then the elements and their spacing will need to move relative to each other. Now I'm trying to find a configuration where the spacing works well for every focal length in the range and I can focus all areas of the image to a nice tack-sharp image... this, it turns out, is not so straight-forward. Lenses start making compromises... some focal lengths are sharper than others, some focal ratios and some focal lengths work better than others, and different areas of the image are better than others (almost always depending on how far that point is from the center of the image.) Also... the all-purpose lenses always tend to have variable focal ratios. This means they don't particularly gather a lot of light and they may not be particularly good at give you an out-of-focus background with quality bokeh.
You can get f/2.8 zooms which perform pretty well, but those tend to be less ambitious about the zoom length (and you'll notice a trend... the more ambitious the lens is with the zoom range, usually the more they have to compromise on the optical quality.)
To solve this problem, manufacturers make cameras that allow you to remove one lens and attach another. DSLRs are perfect for this. You can just grab the lens that's optimized for the task at hand.
SO... if you buy a DSLR, you're getting a physically larger camera -- there's definitely no "convenience" factor here. But you're probably giving up the convenience of a small compact camera so that you can have higher quality images. But to get those higher quality images requires high quality glass. And there's no such thing as high-quality and "all purpose" in the same lens. It's one or the other. So when someone says they want a DSLR with an all-purpose lens, I usually think they probably should have just bought a point & shoot because they're not going to get to take advantage of a lot of the unique benefits of own a DSLR unless they're willing to buy a few lenses that are particularly good at certain things.
As for bodies...
I own a II and I am planning to get a III, but based on your usage it sounds like you might be quite content with the II. The biggest difference is the focusing system. There are lots of other subtle differences as well. Certain the III is the better camera, but it's about double the price and it's only worth it if you plan to exploit the features that set these cameras apart.
Full frame cameras do want better glass. It's worthwhile to "invest" in quality lenses. Here's why:
You'll notice that as you get farther away from the center, the more the quality degrades. Objects in the corners of your image are generally never as sharp as objects in the center. If you look up Canon's lenses on their website, you'll notice that at the bottom of the page they usually include a sample photo by the lens as well as some graphs. Those are the "MTF curve" graphs. MTF = modulation transfer function. Basically it's like an eye-chart for a lens. In the same way that eye-charts have rows of letters where each row gets smaller than the previous row, the chart has sets of parallel lines and each set of lines gets finer (both in line width and the space between the lines) than the last. They're trying to figure out just how fine it can get and still have the lens able to resolve them as separate lines. At some point they'll just blur together. They score the lens based on the finest level of detail it could achieve with the target at center, then at 1mm from center, then at 2mm from center, and so on.
The general trend is that the farther the test target section is from the center of the lens, the less detail it'll be able to resolve.
When you use a crop-frame camera (such as the 550D with the APS-C size sensor) the distance from the center of the image to the corner of the image is only about 14mm. When you use a full-frame camera the distance is greater (because the sensor is larger) and the radius goes out to about 22mm (about 50% farther). That means if you assume the image quality will continue to degrade as you get farther from the center of the axis then you've got a lot more distance to cover... you can assume that the corners will look even worse.
To reduce this problem, you don't necessarily want to use just any junk-quality lens. It's a good idea to "invest" in quality glass if you're going to use a full-frame camera.
