Non-pilot astronauts are not required to become pilots. They fly in the backseat of T-38's partially for transportation purposes, and partially for flight regime training. They are called "mission specialists" and since their job does not require piloting skills, there is no reason to train them as such.
NASA literature says "The T38 is used for
flight readiness training to help the astronauts become adjusted to
the flight environment, including the g-forces experienced on launch" and that non-pilots fly at least four hours per month in a T-38. Pilot astronauts fly at least 15 hours per month to maintain proficiency.
You can look up whether someone has a US Pilot's license using the FAA's Airmen Inquiry Page. A quick search turns up several astronauts who have never been licensed. For example, the mission specialists on the final shuttle mission, Sandra Magnus and Rex Walheim, are not pilots. That being said, I wouldn't be surprised if most, if not all, of the non-pilot astronauts got some stick time in a T-38 since it is a trainer with dual controls.
Obviously the pilot and I'm guessing the commander would be airforce pilots
In the NASA vocabulary, commander is equivalent to what you would think of as pilot or captain, whereas pilot is equivalent to what you would think of as co-pilot. In the Apollo program, the commander landed the lunar module on the moon, not the "Lunar Module Pilot," and in the shuttle program, the commander landed the shuttle, not the pilot.
So, you are correct that both the commander and pilot must be pilots, and some of them did come from the Air Force, although US Navy was more common, and a few came from the Marine Corps. Almost all of them were test pilots.
A few astronauts joined NASA with the qualifications to be pilot-astronauts, but only flew as mission specialists (for example, James Voss and Carl J. Meade). I am honestly not sure whether or not NASA allowed them to fly the T-38's; however, the only photo evidence I can find shows them in the rear seat, which leads me to believe they were not allowed to fly solo.
I have no idea if NASA still uses T-38's
NASA does still use T-38's because NASA still sees a need for pilot astronauts in the future, and the pilots need a way to maintain flying proficiency. In the 2013 Astronaut Candidate Class, four out of the eight were test pilots of some sort, although it's unclear how many will be considered pilot-astronauts.
There is anecdotal evidence that the number of T-38's which NASA operates does appear to be shrinking, which is likely a reflection of a reduced astronaut corps size, budget constraints, and the fact that the aircraft fleet is about 50 years old. According to unconfirmed sources (wikipedia), the number of NASA T-38's is expected to be reduced from 32 to 16 by 2015.
Other Space Shuttle-specific aircraft have been retired entirely, such as the Gulfstream II Shuttle Training Aircraft and the Shuttle Carrier Aircraft (the latter wasn't flown by astronauts).
