Does the classic PlayStation 2 controller support a USB standard? I.E., can I cut the cable and solder a USB connector to it?
Directly interfacing a PlayStation 2 controller to USB is a losing proposition. The protocols and hardware are not at all compatible. You're in for way too much work, risk, and pain.
You can, however, buy a cheap adapter for just a couple of bucks. These tend to be cheap both in terms of cost and in terms of reliability, so I suggest buying someplace with a good warranty/return policy.
You can not simply replace a PlayStation 2 controller connection with a USB connection, and expect it to work.
The immediate clue is in the wires. Not only do the data lines mis-match, but the power line for the controller uses 3.3 volts, and USB supplies 5 volts.
Simply connecting the controller to a powered USB slot could cause unsalvageable damage.
With specific knowledge and skill in electronics and the data being exchanged, it is definitely plausible. There are guides online for doing so, but they can be are generally complex, and not for the layman.