Re: Can you fix a guitar that doesn't play in tune?
Just to be sure, there are two different concepts:  Tuning instability; and intonation.   I'm assuming you are talking INTONATION.
You need to set intonation with new strings that have been stretched.  I like to let them settle for 24 hours, unplayed, before I do any intonation work.  Once they have been used--intonation deteriorates.  Keeping my hands and strings clean--I get about 15-20 hours of gigging play-time from a set of electric strings.  They do wear out. 
If you have frets that have not been leveled and crowned properly, the guitar will not play in tune.
If the nut has not been cut properly, the guitar will not play in tune. 
If the bridge saddles have not been cut properly, the guitar will not play in tune.  Yours look a little suspect to me.
If the neck joint is unstable, or if the the top of a hollow-body or semi-hollow is unstable--well, you get the idea.  If the neck itself is weak, or you have been pulling on the neck to get vibrato--that could be the cause of the problem.  (That kind of vibrato is only good for guys who can afford to replace necks or buy a lot of replacement guitars!)
Oh, and make sure that the tuners aren't loose and the nuts snug but not over-tightened.  There should be zero play in the tuner mounting.
You might try holding the guitar in playing position, like if you stand with the guitar with a strap--and set the bridge intonation with the guitar in that position.  Yes, it will be awkward--perhaps you'll want to enlist the help of a friend.  It can make a difference with some guitars.
You might also try checking the intonation by using a capo at the 1st, 2nd, 3rd fret--and then checking the respective fretted octave; rather than the open strings.
I had a very nice Gibson guitar years ago with not only an intonation problem but the tuning was unstable too.  I finally sold it, and was glad to be rid of it.  You'll have to figure out how much more money you're willing to spend on this guitar.  And you might just be sensitive to the evils of Equal Temperment Tuning--and only a guitar with a Buzz Feiten or Earvana modification will satisfy your golden ears.
Good luck.
Bill