Humphreys & Larsen (1995) suggest, using star count information, a distance of $20.5 pm 3.5$ pc above the Galactic plane; consistent with, but more precise than the Bahcall paper referred to by Schleis.
Joshi (2007) is more guarded, investigating some systematic uncertainties in the estimation techniques and ends up with distances between 13 and 28 pc above the plane.
This paper gives an excellent review of the topic in its first couple of pages.
The Sun moves at about 15-20 km/s with respect to a local standard of rest defined by the general motion of stars in our vicinity around the Galaxy. In three-dimensions, this "peculiar velocity" is $U=10.00 pm 0.36$ km/s (radially inwards), $V=5.25 pm 0.62$ km/s (in the direction of Galactic rotation) and $W=7.17 pm 0.38$ km/s (up and out of the plane). (Dehnen & Binney 1998)
The Sun executes oscillations around its mean orbit in the Galaxy, periodically crossing the Galactic plane. I borrowed this illustration (not to scale!) from http://www.visioninconsciousness.org/Science_B08.htm to show this oscillatory motion.
As the Sun is currently above the plane and moving upwards, and each cycle takes about 70 million years with an amplitude of 100pc (Matese et al. 1995), it will be roughly 30 million years before we cross the plane again.
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