With a false neutral does the neutral light come on and how do you shift out of a false neutral?

no, and you just just hit the gear selector up or down back into whatever gear you want.

--EDIT

You'll know it when it happens, you'll go to get back on the throttle and just hear *WEEEEEEEEEEEEEEEEEEEEEEEE* of the motor reving up and you won't go anywhere....how embarassing!  

Mine has done that a couple times but once it kicked into gear before the the engine down revved.  I thought the back tire came off or the engine fell off the bike and I ran over it.

yeah i get false neutrals on my 2002 all the time.  what causes it? how do you get rid of it?
When I shift it in the gear it makes a bad click sound even if i have the clutch pulled in.  i don't like it

ya false neutrals are caused by lazy shifts....you don't quite pull the shift lever all the way up and the dog bones sort of guess what to do....when they don't well then you get a neutral....Not sure if the green neutral light comes on or not...I would guess probably not but it could....Like someone else said it's pretty embarassing and I didn't exactly look down at my instruments to see if it was in neutral I was looking around to see if anyone was staring at me...

snowey-is that harsh click what you experience at a stop light? or being motionless and then kicking it into gear? if so I had that trouble too and I think it's just because of the "clunkliness" of the GS. double check your idle and clutch adjustment to see if that helps and the oil level...if it get low the "clunkliness" is even louder. AFter awhile I found double cultching solved the take off gear problem but I soon after replaced the clutch disks and cable and the problem dissapeared.

My light comes on when I've done it.  (the numerous times)

But I catch it real fast. I'm like.. woah, not going faster... click..
The first time I started looking all around.. like the guy on the sidewalk was causing it  :lol:

the clunkiness is after i try to accelerate and i am not in gear. I then quickly pull in the clutch and drop it into gear. at which point i get a clunk