I assume it's more difficult to get a sharper picture at F2.8.(?)
And when selecting Full Auto, shouldn't the camera come close to giving me a sharp shot?
1. The amount of the photo in focus (Depth of Field) is less at f/2.8 than at smaller apertures. So you have less room for error -- a very slight focusing error is much worse at f/2.8 than at f/8. But when properly focused, the area in focus will be very sharp even at f/2.8.
2. No, absolutely not. The camera doesn't have any idea what part of the image you want to be in sharp focus. For example, in your macro flower shot, you could want the tip of the stamens in focus (or maybe those are the pistils -- biology was a long time ago), or maybe you want the petal, or the bee hovering over the flower. On full auto mode, the camera will simply focus on whatever part of the subject has the most contrast (or is the closest to the camera, or whatever algorithm the camera uses.)
In each of your examples, the shutter speed is too slow to guarantee a sharp photo hand held. The camera moves during the exposure, and results in slightly blurry images. In those situation, you can either boost your ISO to get a higher shutter speed, use a tripod, or you can try shooting two or three images on Continuous High motor drive -- this often results in one photo being noticeably sharper than the others. (I get a lot of sharp photos using this last technique, at surprisingly slow shutter speeds.)
This is not to say that you don't have a technical problem with your camera and lens. I have a 40D and it absolutely would not focus my 24/1.4 with any precision. After testing it very carefully on a tripod, with good technique, I sent the camera, lens, and some test photos to Canon, who fixed the problem. (Newer cameras allow the user to fix this, for the most part.) But my 24-70 is a great lens on the 40D, very sharp and makes nice photos.
So your best bet is to test the camera and lens combination on a solid tripod, using autofocus (choosing the center point), in various kinds of light, and check the resulting images. I also like to manually focus using Live View, zoomed in to 10x to adjust the focus. Then I compare the two. Always defocus the lens before initiating auto focus. Give it two or three tries, and check all of them. If you are still having problems, call Canon's service department. They have fixed this issue for me under warranty when the camera and/or lens was new.
In the meantime, tighten up your shooting technique. Make sure your shutter speed is fast enough, press the shutter button slowly and carefully, brace your arms in against your torso, etc. Use a tripod when necessary. Good luck.