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The trem should be flush or just proud of the body, in most cases. If it's too low, loosen the tension on the strings, and unscrew the trem posts a bit, until the height is correct.
If it's not holding a tune, you may need to increase spring tension a bit by turning in the screws which attach the spring claw on the underside. I would leave this until everything else is set up, though.
You can't get low action without fretting out if there is significant bowing in the neck, so you really need to get this right first. Truss rod adjustments aren't a big deal, as long as you work slowly and carefully. I use a two foot straightedge on the neck, to ascertain how much relief the neck is set with. I look for gaps under the 4th to 10th fret. If I see light, there is some relief, so I tighten the truss rod a bit until there is just a hair of relief, or none at all. Tightening or loosening, go no more than a 1/4 turn at a time (loosen the strings first), and give it a moment to settle.
Do the neck first, then adjust the height of the trem, then deal with string height. Fretboard radius isn't an issue, as each string's saddle has a height adjustment. So you can easily adjust each saddle until the string height is correct for you.
Check and adjust intonation once the rest is done, again by adjusting the saddles closer or further from the neck, as required.
Once all of this is done, if you're still going out of tune, increase the trem spring tension, retune, and try again. Some back and forth here is inevitable, the key being to finding the balance point where it does hold tune.
Hope this helps. I have numerous guitars with trems, and they do take some fussing to set up properly, but once done, I have zero tuning issues. Many resources on the web for adjusting trems - just google what you need.
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