For example, E Melodic Minor would have C#, allowing for a ii7-V-I cadence.
Yes and no.
"Yes" because there are certainly moments where he uses the raised sixth and seventh scale degrees of a minor scale. In fact, a recent question discussed such an example: Functional analysis of chorale 'Wie wunderbarlich ist doch diese Strafe' BWV 244/46
But it's also "no" because there really aren't "melodic minor keys." Minor is just minor, and the fact is that music (especially by someone like bach) rarely sticks to one "form" of minor. In practice, music in minor flows freely between the natural, harmonic, and minor forms of the minor scale. So it's a bit of a misnomer to ask whether Bach ever used a "melodic minor key"; instead, we simply acknowledge that music is in a minor key, and scale-degrees six and seven have a particular fluidity that allows them to be adjusted at various points.
Yes, all the time. Because Bach (although he was quite prepared to be 'modal' at times) set down the foundations of functional harmony, built on dominant-tonic resolutions. And if you want a perfect cadence in a minor key, you need a major dominant chord including the leading note. And, for smooth voice-leading, that implies a melodic minor scale. ('Scale', not 'key'. 'Harmonic minor' isn't a key.)
The raised 6ths and 7ths in minor are mostly used when the line is ascending toward the tonic. Bach used this very often. When the line is descending away from the tonic, the 6th and 7th degrees are often used as they would be in the natural minor. It is quite common to have both the natural and raised 6th and 7th degrees in the same phrase, depending on how the line flows. One line can easily go up and down in relation to the tonic within the same phrase, and the notes would be adjusted accordingly.