For the first part, you will work the first 27 stitches of your row with the two strands held together as you've been doing, and after the first 27 stitches you will stop working with those 2 strands. Leave that yarn attached because you still need it. Now, take another 2 strands held together and with those two strands you will work the next 8 stitches. In order to do that you can just start knitting with it - the first few stitches will seem VERY loose but that's just because the end hasn't been woven in yet. You can do that now or later.
The 8 stitches that you just worked will be put on a holder now, but you will continue with the 2 strands held together that you just used for the center 8 stitches, to work the last 27 stitches.
So now, you have working yarn coming from BOTH sides of the center 8 stitches. You will use one working yarn (2 strands) for one side of the neck and the other working yarn (2 strands) for the other side. Both sides of the neck can be worked simultaneously by alternating the balls of yarn when you switch from side to side.
When it says to BO 4 stitches, twice is means that you will bind off 4 stitches at the neck edge on 2 rows, NOT 8 stitches all at once. Binding off can only be done at the beginnings of rows, so when you are at the neck edge of both the left and right neck edges you will bind off 4 stitches on 2 seperate rows.