Can an ATX PSU power a USB port directly?
The 5V rail will most definitely be able to supply the power that you need in your setup. For most ATX PSUs, the +5v rail has the largest current capacity of all rails on the supply. Many, if not most devices will charge as soon as there is a voltage applied to their USB port, so I'd say that aspect of your setup is also good. I wouldn't worry about a device overdraw; the current control is generally built into the device rather than the charger.

The only thing to really consider is similar to what you mentioned in your question; some devices will begin to charge at say, half of their current capacity, and only ramp up to full current once they've received some data from the charger/host device. But, this is only really an inconvenience, and most devices will charge with at least some initial current. The only device I can think of off the top of my head that won't charge at all without this enumeration is a Sony PlayStation 3 controller (but I'm sure there are more).

The final point is that this really just varies on a device to device basis. Some devices will charge with max current right off the bat, a few will be limited to initial max current, and a couple might not charge at all.