#1
Is there any way to eliminate the picking noise(the slight ticking) and the white noise that is coming from my amp when I am not playing? I have a BOSS noise supressor pedal but I guess it doesn't work for that or I just am not setting it up right.
#2
While I have no experience with noise suppressor pedals, I think they can only work on the noise generated by cables, pickups and pedals. In regards to the amp noise, I don't think you could get rid off it with said pedal because its acting before the noise is there.

(if that makes sense)
My "Rig":

Fender American Telecaster
Boss OD-2
EHX Small Clone
EHX SMM avec HAZARAI
#4
Pure_Morning is somewhat right. A noise suppressor will only kill the noise that is below the threshold setting you have chosen in the line of effects you have running to it.

Every amp generates some white noise, some might call it a light static or hiss, from your amp, it's gonna happen especially when you're on the high gain channel. It should not be evident if you were playing with a band, but if it's too loud then you want to start eliminating pieces of equipment to see if something else is causing the loud hiss. Try a process of elimination. Switch out cables, try a different guitar or simply turn it on with out plugging anything into it to test it out. If it is your your amp, then you may want to take it to a repair shop.

As for the picking noise; is it really noticable when playing on higher octives? A noise surpressor will not eliminate this either, try cutting back on the treble or the gain or the tone knob (if you have one) or try different distortion pedals. I usally go with turning down the treble.