Re: Tuning Issue
Your strings are sticking somewhere. When you pick them, the resultant vibration loosens the constriction that's holding them, the tension along the entire string length equalizes, and they go sharp/flat/whatever. The "jumping" in pitch when you tune the A string also indicates that the string's sticking, then releasing as you increase the tension on the part of the string between nut and tuners, by continuing to rotate the peg.
Most likely place is still the nut slots. The graphite lubrication may not have been adequate to overcome the problem, and the slots may need to be re-cut. That's a job for a pro.
Bridge saddle slots are seldom deep enough to grab the string, though it's not unheard of.
There also could be a tailpiece issue, as Jim details above. I've not heard of string loop windings getting trapped inside the tailpiece, but it's surely a possibility.
This is the point where I take my instruments to an experienced repair person. You've tried a number of expedients that haven't solved the issue, so perhaps a "wiser head" should be consulted?
Allen Hopkins
Gibsn: '54 F5 3pt F2 A-N Custm K1 m'cello
Natl Triolian Dobro mando
Victoria b-back Merrill alumnm b-back
H-O mandolinetto
Stradolin Vega banjolin
Sobell'dola Washburn b-back'dola
Eastmn: 615'dola 805 m'cello
Flatiron 3K OM
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