Hi and welcome!
This sweater is knit in stockinette or stocking stitch so the knit rows are the right side and the purl rows, the wrong side. Step 6 should be a purl row on the wrong side and you should now be ready to begin a knit row on the right side.
Second (main) question: Note: All Leg sections are worked at the same time using separate balls of yarn for each section.
I am guessing this refers to the leg edging, but really don't have much of a clue. After the stitches are cast off in step 1, step 2 says to beg with a purl row. Does this mean to work on the 56 stitches that remained after casting off?
For the large size you begin on the knit row and knit 7, cast off 6, knit 56, cast off 6 and knit to the end so you have two areas of cast off sts and 3sections of sts on the needle. Now you're ready to purl back across to the leg opening, drop the working yarn, attach a new ball of yarn and work the middle 56sts, drop the working yarn, attach a new ball or end of yarn and purl the last 7sts.
Third question: Leg edging - so of course there are two leg openings. It instructs, "with first double pointed needle, "pick up and knit 7". What does pick up mean exactly? Is this where I use the "separate ball of yarn"? And then a second double pointed needle is to "pick up and knit 10". Is this second needle for the second leg opening, or is it for the opposite side of the opening?
There's a video for picking up
sts under the Free Videos at the top of the page. You'll put new sts onto the needle or pick up sts around one leg opening first and when that's finished do the other leg opening. Don't worry about the finishing steps for the moment. You can ask again about this when you get there.