Thanks, mikeluke.
I think I'm just going to go for the "five springs" approach to get the trem flush against the body and see how that goes.
The tuners actually seem pretty decent, so I'll keep them for now. If I do decide to change them I'll probably go for locking units: I love 'em, and think they should be used even on non-trem guitars.
You're right about the electrics - my five-way is shot to hell! I'm thinking of ordering a completely new, pre-wired scratchplate so I get quality pots too. Which leaves me with a problemette: normal Strat scratchplates have 11 screw holes, whereas my Squier only has 8. Are the screws self-tapping and should I just screw the thing down regardless, or should I drill some holes first?