Need help understanding this op amp/BJT configuration to troubleshoot why I'm reading different voltages than expected
a) You'll have a higher output current capability with this configuration compared to the one without the op amp alone.

b) I don't know if this configuration has a specific name. I would call it "A BJT with current limitation"

c) Are you sure about R7/R5 and R9/R4 combinations? Those look like gains of 4. I'm guessing that the op amp can't output more (less) than a defined voltage (which is lesser than +15V or higher than -15V depending on which op amp you're looking at), not high/low enough to output the full +15V/-15V of your rail.
That being said, you should make R7=R5 and 2xR4=R9 to get the targeted gain of 2.

Regarding the diode voltage drop you talked about, all I'm seeing on the simulation is an oscillating voltage (Vpp=11.3V and Vdc=-). Let us know if the problem persists after the gain correction to think about it.