Here's my exact process when handling quad lines, using a single LaserPro winder...
Setting up:
Handles already attached, I hook them onto the ground stake in a forward-flight position.
Walk downwind, unwinding to the end.
Separate R-L pairs.
Set kite leading edge DOWN.
Attach right pair.
Attach left pair.
Draw lines and kite taut, so line tension keeps it flying into the ground (safe).
Walk to handles, draw all lines tight to cancel fake wraps.
Remove 2-3 twists (normal) by moving the handles around each other, and fly.
Packing up:
Land kite leading edge DOWN with no twists in the line.
Hook handles onto stake in forward-flight position, so kite is secure.
Walk to kite.
Remove left pair, larks head top line onto the bottom line near the knot.
Remove right pair, larks head top line onto the bottom line near the knot.
When looking at the pairs, you can treat them as two lines (two bottom loops together) and start winding.
As you approach the handles (staked), wind DOWN to them to make sure they don't spin around or cross.
When all the line is wound up, fold the handles over the winder and strap it all together (bunji, velcro, rubber band, whatever).
Three key items...
By larks heading the top to bottom at my kite, I've drawn the bottom of my handles out so that they fold over the winder easier.
If I do remove my handles once I've wound up my lines, I then larks head the BOTTOM to the TOP (opposite of the kite end), so there's no undue slack in them.
Never try to take out what you think are wraps until you've gone to the handles and pulled everything taut... If you do have wraps and can't figure out how they go, leave everything attached, walk down to the wraps, stick your fingers in and slide it up to your handles where you can see them easier.
Done this way, you shouldn't ever have to disconnect a line to take out wraps... Everything should come out just by rotating the handles together, separately, or passing one handle through another... Again, if done correctly, you won't even have to do much of this.
In 16 years of flying, I haven't found ANY difference between figure-8 or straight winding (I wind straight)... The only exception is when I'm using larger lines (150# or 200#) that tend to fall off the winder... In this case, I put in 20 straight winds, one figure-8 to lock it down, 20 straight, one figure 8, and so on.
That's pretty much it.