From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?
It will never reach such a high velocity. The moon is drifting further from the earth due to tidal acceleration. This process is, at the same time, slowing the rotation of the earth. Once the earth's rotational period matches the moon's orbital period, the earth-moon system will be tidally locked to each other (note: the moon is already tidally locked to the earth), and the acceleration will cease.
To briefly explain the mechanism, the gravitational pull between the earth and the moon causes tidal "bulges" to extend out on both bodies (just like the ocean tides, except that the entire surface moves slightly, not just the water). Since the earth rotates faster than the moon completes one orbit, the bulge on the earth lies slightly ahead of the earth-moon line, because the earth is rotating so quickly. This bulge gravitationally pulls on the moon, speeding it up, while at the same time the moon pulls on the bulge, creating a torque on the earth and slowing down its rotation.
The moon's recession from the Earth is driven by tidal interaction, and while the rate varies, it is slow and gets less on average over geological time. It should stop receding in tens of billion of years, long before orbital radius reaches edge of the Hill sphere, where the Sun's gravitational influence equals the Earth's and allow the moon to wander away from the Earth. The precise timing of this and the eventual fate of its orbit will be determined by other events. The rate of lunar recession depends on the rotation of the Earth, the distance of the moon, the mass of the oceans, the position of the continents, the mass and extent of the atmosphere, the mass and fluidity of molten rock below the crust, and possibly other things. As long as there is tidal drag on the Earth (from the Sun as well as the Moon), its day will increase and eventually reach the period of the moon's orbit. The Wikipedia article reports that this would happen in about 50 billion years, at which time the period of the Moon's orbit would be 47 days (assuming the Earth and Moon survive the Sun's "red giant" phase, about five billion years from now). If solar tides are still operating then (the Sun would be a black dwarf, about half it's present mass), the Earth's spin would continue to slow down to less than lunar orbital rate. The Moon's orbit would then decrease in size until it merged with the Earth, many, many billions of years later. Long before any of that happens, however, our descendants may have engineered the inner solar system to suit themselves.