PNP led on when it should be off
I would suggest you try placing the R1 immediately next to the base of Q1 so that it is between its base and emitter. That way it keeps the Q1 turned off if there is a small current going through R2.
  I would also suggest that you increase the value of R3 to anywhere from 220Ω to 680Ω (depending on LED's forward voltage and the total voltage applied, which I believe is around 2V and 5V respectively, as well as on the type of LED - is it superbright?); you could also reduce R2 to 10kΩ so that the transistor is more likely to work as a switch and not in its linear region.