The low 'E' pin is a bit high, too. Probably nothing; maybe the holes have been reamed differently (although that would surprise me of a Yamaha...). The holes are actually tapered. Next time you re-string, try swapping the 'D' pin for the high 'E', for instance..?
As for the tuners, they're very rarely an issue, with any guitar. Most of the time it's just a question of how to put the strings on. Here's what we do at our house...
I'll assume that the bridge pin is now firmly holding the ball end in place, as described above ^^. I thread the string through the hole in the tuner. For the skinnier strings, I loop around and pass through the same hole again. In doing this, I keep some slack in the string, enough to hold it about an inch or so above the fretboard. I'll wind the tuner, keeping the string taut above the fretboard until it's settled into the nut slot. My first turn, I'll guide the string to below the tuner hole. The next turns I'll guide progressively up the tuner post. There should be about three or four turns, if the amount of slack has been guessed correctly. Repeat for the other strings.
Once all the strings are fitted, I'll tune up to pitch (I start with the 'A', then tune 'em all relative to that...). Once they're all up to pitch, I'll seize each string at its mid-point, and gently lift the guitar up, solely by the string. There's no fear of the string breaking in doing this; it settles them and tightens up any slack anywhere, just by the guitar's own weight. Once I've done this for each string, I'll tune up again, more precisely. This way, new strings are pretty much bedded in from the outset. Maybe a tweak or so the next day, but all will be stable from then on, until the strings wear out through aging or intensive playing.
If there's anything not clear in my description, feel free to ask for more. If you have your own method that suits you, carry on, of course..!