Soldering questions - recovering data directly from eMMC via microSD adapter
You have embarked on a challenging task.


You pick the correct wire size for the job, 38AWG, it will not break pads. But still be very careful and do not bend wires, secure them with capton tape.
You don't need to strip ends. All you need is a slightly hotter iron tip, 250-300C. Using a solder with resin core, just touch the solder tip with solder wire, and while the blob still has melting resin core, stick the wire into it. The insulation will gently burn out, and you will have a perfectly wet wire tip.
Voltage rails should be connected. By SD/MMC specification, memory chip must be tolerant to default 3.3V, so it should work, although less power efficient. However, I am not sure if you can skip DAT1-DAT3. The SD controller will first access the eMMC in one-wire mode, which will tell the controller that the chip is 8-bit wide. The SD host will then try the connection at advertised frequency and bus width. I am not sure if your SD host will fall back into one-wire mode if it has no right response. 
Not sure what do you mean here about the balls. The wire is either soldered, or not to the spot.


Having done this kind of workaround myself many times, I would not try this without a professional grade soldering iron with 0.1-0.2mm tip, and a good stereo microscope.