Why do we use pull up resistor with PIC?
A floating output can be due to the pin being configured as an input (quite common) or it is truly tri-state


  1.How will the floating output from the PIC affect the base of the transistor? Will it make it fluctuate between ON/OFF state?


With the pullup in place, the transistor is solidly pulled to the OFF state.


  2.What would happen if we don't have the pull up resistor?


Without the pull up, the state of the transistor will be indeterminate; there will only be leakage (in the order of μA) in or out of the controller pin, so the transistor might be either on or off, or possibly fluctuate as you mention. 


  3.Is there any risk of the transistor being destroyed because of the floating output form the PIC?


Highly unlikely, but it might depend on the transistor load.