Hi @enigmatic319.
Are you sure that the tremolo has been setup correctly, and that the strings were stretched sufficiently before locking the nut? This accounts for most of the issues with tuning stability with Floyd Rose type bridges.
It would also be helpful if you could provide the make and model of the guitar, and whether it was purchased new or used. Also, the string gauge on this guitar and the string gauge you’re used to playing on.
If the bridge is correctly set up and in full working order, then it sounds to me like the issues you’re having are the result of poor technique.
Regarding your points:
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For most people, in most playing positions, the picking hand doesn’t truly “rest” on the bridge. Gravity pulls the hand down, not horizontally into the bridge. If the bridge is pulling sharp, you’re probably applying force into the body of the guitar. I noticed I had this habit when I first started using floating vibrato systems.
It not necessary to exert force into the guitar with the picking hand. String damping only requires that you lightly touch the strings. Any benefit to stability you might feel from a firmly anchored picking hand is hugely outweighed by the the looseness and mobility that playing without an anchor allows.
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With a Floyd Rose type vibrato system, hitting the strings too hard causes the bridge to move and the strings to ring out of tune.
Why do you feel the need to pick that hard?
Can you pick more softly and still play confidently? Do you have issues with string noise if you try to pick softly?
A hard, percussive attack can be achieved by picking more lightly if that’s what you’re trying to achieve. Notes actually have less sustain when you pick with too much force, and in my experience, that amount of force is still much less than the force required for the strings to ring out of tune.
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After playing open chords and checking the tuning of the open strings, are they still in tune? if not, your Floyd Rose is poorly setup (or faulty) or your strings are not sufficiently stretched before locking the nut.
If the open strings are in tune and you perceive the chords to be out of tune, you’re most likely fretting too hard and causing the fretted strings to bend sharp, which causes the open strings to flatten.
Many Floyd Rose equipped guitars have jumbo frets, and it’s not unusual for players who are used to feeling the fretboard under their fingers to fret too hard.
I actually think most guitar players fret much too hard; you only need sufficient force for the fretted note to ring out clearly, anything more is unnecessary, and in fact detrimental. Most players who I have let play my guitars then complained that the guitars “kept going out of tune,” even though they were in perfect tune when I checked them.
I could be wrong in my understanding of the situation. It could be that there is some issue with the bridge or the setup which is exaggerating these issues. You haven’t given me much to draw conclusions from.
If I’m right, I’d suggest you keep the guitar and try to develop a lighter touch.
