No - DON'T CUT the yarn!!
Work those 4 stitches back and forth like you were knitting a scarf - a lit-tle tiny scarf, till however long you want the strap to be, 5 1/2" per your directions.
Don't understand about how the bound-off stitches unravel when you try to knit the 4 sts, but maybe you could pull up a loop and feed your ball of yarn through the loop to make a sort of knot? Honest, it's vital to have those 4 (or 5 or however many you like, if you want a wide strap) stitches to knit over.
Okay, on to the seed stitch. For 4 stitches:
Row 1: Knit 1, Purl 1, Knit 1, Purl 1. Turn.
Row 2: Purl 1, Knit 1, Purl 1, Knit 1. Turn.
Repeat these two rows exactly for the seed stitch. You'll get little bumps or "seeds" instead of the usual ribbing that k1,p1 will give you. In essence, you're purling the knits and knitting the purls.
Hope this helps,