For the pattern you linked, you will cast on your stitches with two strands of the main colour, and then right away with the first row you work you will be using the contrast colour. A double strand is used throughout so when you start your constrast colour you will start with 2 strands held together.
As far as the short rows, really you just need to go row by row following the instructions given and turn your work around when it says to, even though you have not worked all the stitches of the row. You are working part rows to create the shaping at the crown of the hat. Because you're working the hat sideways, short rows are used. If you were working the hat top down or bottom up, you would be using increases or decreases.
Regarding the trinity stitch where you have to (K1P1K1) in the same stitch. You will knit the stitch and before you slip the original stitch off the left needle you will bring your working yarn forward and purl the same stitch. Return the working yarn to the back and knit the same stitch once more. Now, slip the original stitch from the left needle. You have created 3 stitches on the right needle.