Regarding your AF question on the 60D - I'm a wee bit confused on the issue you have with half-pressing the shutter to AF. As far as I know, ALL cameras, both P&S and DSLR, have the half-press indent to autofocus before fully pressing to shoot. There are some cameras that have a second button on the camera that allows you to autofocus separately from the metering...if that's what you are referring to...though I'd be very surprised if the 60D didn't offer a similar capability somewhere in the button arrangement or customization. Honestly though, if you're an amateur shooter, it would not likely come up often if at all that you would need to separate the metering and focus operations, and getting to know how to use half-press focus and metering, or when to use AE lock or AF lock separately, would be a high priority that would really help you.
As far as other cameras to consider - I think it would behoove you to look at, and more importantly, handle if possible, some other models that could very likely do everything you need them to do and more, for less than your budget. To be very honest, even the most entry level camera can handle all the types of photography you mentioned, and you likely won't get better photos from any of them with a few rare exceptions until your skill exceeds the camera's controls. 'Better' cameras aren't really about getting better images, as much as they are about getting better control, and providing more tools to make getting hard images easier.
I'd recommend taking a look at the following: Nikon D3100, Pentax KX, Sony A500/550 on the low side of your budget - plenty of room for a nice lens or a few used lenses. I'd recommend looking at the Pentax KR, Sony A580, and Sony A55 if you don't mind sliding up to around the $800 mark on your budget...you'll get some nice improvements and some excellent sensor performance. Or the D7000, 60D, or Pentax K5 if you don't mind bumping your budget up over the $1K mark.
The A580 is worth a strong look because it shares the same 16MP sensor that the more expensive D7000 & K5 use, but for $800 or under. It is fast, nice grip, solid performer, excellent high ISO, has the best live view system ever put on a DSLR, can shoot 5fps for sports with focus, or 7fps with fixed focus, has in-body stabilization, a tilt screen, and a cheap available backlog of lenses.
The Pentax KX was a real price leader - under $500, and packs a lot of camera for the budget, with good high ISO performance and a fast 4.7fps shooting.
Nikon and Canon are excellent and well known, and you won't go wrong with either. The models you are considering are mid-range and pricier, with nice build and features. Some advanced entry models will shoot as well, and for a lot less money, still likely delivering everything you'll need for years to come. And Olympus, Sony, and Pentax also make excellent cameras that can meet all your needs, and are worth a look...they all deliver some nice features for a great price, some things unique to them - what feels good in your hand, fits your budget, and has the features you need is the way to go, whatever brand name it has.
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses