You can get a Strat with a floating Trem, but most only allow you to lower the pitch. A floating trem is one you can push down and pull up on.....

As for the LP for shredding, take this into consideration. If your looking for a guitar to shred with, look for a guitar that's easy to reach the higher notes. Most shredders also prefer a 2 octave neck scale (24 frets). But, it still all depends on your personal preference, whether you like a fat or skinny neck (a skinny neck is also a preference of shredders, they feel "faster). I'd pick a LP over anything, but to me, a LP is not a shredders guitar.....