#1
The string height on my guitar is at 3.5mm at the 12th fret and I was wondering if that is too high and if the height can be lowered by dropping the tremolo.
#2
It's way too high for *me,* anyway.
Assuming that you have level frets and that your nut is cut properly, you should be able to get action that's half that or less.

The nut and the bridge (in your case the trem) are the two points of string suspension, and at least one manufacturer used to advertise action "as low as 1/16th" at the 24th fret (!) with no buzzing frets." You're double that at the 12th.

Aside from a properly cut nut and level frets, you'll want to make sure that you have minimal "relief" or curvature in the fretboard (the *best* way to measure it is with an 18" straight edge and a set of feeler gauges) and then lower away.
#3
It is a lot higher than I would have it. I have 1.6 to 1.8 mm treble side, about 2.4 mm bass side, and many electric players would go lower than that. Before you adjust the trem or saddles, make sure that the neck relief is right. If you press down on the 1st and heel frets, the gap between the string and the 6th fret should be about the thickness of an business card. Credit card thickness would be too much.
#4
I run mine between 1.25mm and 1.5mm on the bass side.  All that really matters is that you are happy with it.  If you're not happy with it listen to dspellman 
Guitars:
Jackson Kelly KE3 - MIJ (Distortion/Jazz)
Jackson DKMGT Dinky (EMG 81/85)
ESP E-II Eclipse Custom (JB/'59)
ESP LTD EC-1001FR (EMG 81/60)
Fender MIM Strat

Amps:
Mesa/Boogie Dual Rectifier Roadster 212
Laney IronHeart IRT-Studio
Peavey Vypyr 30
Peavey ReValver Amp Sims
TOOOO many T.C. Electronic Pedals. . .
#5
dspellman 

I think it might have been Washburn that advertised that action height thing. Some may not know this, but with careful setting of the neck relief, you can get the action lower at the highest fret than at, say, the heel fret on many guitars.

I prefer the use the string under tension rather than a straight edge, nothing gets much straighter than that.
#6
It's way too high. How you adjust it depends on what sort of tremelo you have and if it's minimum setting is OK for you and if you can take the bridges down and what shape the neck is in. Really, if you don't understand it take it to a pro guitar tech and get it fixed.
#7
Quote by Tony Done
dspellman 

I prefer the use the string under tension rather than a straight edge, nothing gets much straighter than that.


Using the string is a "rule of thumb" thing. And you never want to use one of the wound strings; they can easily NOT be straight (and if you're holding down the first fret with a capo, it probably won't be). This is especially true if you're using new strings just out of the package; they can carry a slight bend (in any direction), depending on the string make, stiffness, etc. I've seen similar "rule of thumb" measurements as suggesting a business card, a playing card, even a credit card. The original rule of thumb suggested a "new playing card," which is fairly standardized, but a used playing card can be much thicker (due to moisture absorption, etc, paper fiber expansion and bending, etc. etc.,). And finally, the same rule of thumb has variously suggested that you want the card to 'barely' raise the string, that you want to be able to pass it under the string with it touching but NOT raising the string, and so on. With the aluminum straight edge, there's little question about whether the feeler gauge can get underneath (and no more).

A lot of guitarists will claim they do the whole setup by "feel," including setting the action height, checking the frets for level and a whole lot else. As long as they're comfortable with the results, that's fine.