#2
Um... there'd be nothing to drop the bridge into. That's why there's a hole in the body, for the trem.

And what do you mean screw in the claw? You only screw it in enough to balance the tension, Haven't you ever used a guitar with a trem?
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#3
With a body like that, you probably would not be able to use a hardtail bridge. Most hardtail bridges require their mounting screws in the same location has the vibrato rout is. It's easy to lock down a vintage vibrato bridge anyway, you just screw the six pivot screws all the way down, add on all five springs and screw in the spring claw as far as it will go before things start to buckle. It doesn't have to be precise and you can always fill in the gaps aroudn the sustain block with extra chunks of wood to keep it sturdy. The added benefit of doing it this way rather than trying to fit a hardtail bridge is you'll have a solid metal block runing through the guitar which adds to the sustain of the guitar significantly.
If you really want a hardtail Strat body with that kind of routing, use Warmoth. They'll make you a body pre-drilled for whatever bridge you want.

As for the tuners, most tuners fit 10mm peg holes, or include bushings to fit 10mm holes. The easiest thing though woudl be Planet Waves Auto-Trim tuners; they're made for 10mm post holes, they're locking, they've got an 18:1 gear ratio and they cut the excess string off for you too.
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