I made one of these several years ago and later I made a couple of DK hats in the round from the top down, one had earflaps, but that is the total of my experience with DK. But I’ve been thinking about doing some more of it and have been brushing up a bit and learning some improvements.
Tonight I started this project again but not exactly as it is given. I’ll tell you the changes I made. But first…
start at bottom right corner
row 1 and odd numbered rows are read from right to left
knit the stitch in the colour shown and purl the opposite colour. so for a red square i knit with the red yarn first then i purl the white yarn
even numbered rows are read from left to right
knit the stitch in opposite colour to the square on the chart
so for a red square i knit with white yarn first then purl with red yarn
when knit stitches have both yarns in back and for purling have both yarns in front
Your understanding is correct. 
the instructions say to cast on 30 but i only count 29 squares
I’ve noticed this several times. I can’t really account for it, and don’t remember what I did the first time I did this. It is possible that with the purl the last 2tog and slip the first one technique that she employs you need that extra stitch for it to work out. :shrug: But this is where I veered from the pattern slightly.
I learned from a guy who calls himself Fallingblox (I think that is the spelling) on Ravelry (who is quite an expert at DK) a different way to handle the edge stitches that I wanted to try. But he points out that you need to have a good border of solid color at each edge to make it work. So I decided to add one stitch of white to each side of the chart as it is written, which makes 31 stitches on each side of the potholder, for a total cast on of 62 to work the project. (Forget what they said about 30, you only need exactly the number the chart shows for this method, but I am adding one stitch to the chart on each side. So it ends up with 2 edge stitches on each side in white on the front and 2 stitches of red on the back side on each side.)
Then at the beginning of each row you just work the first st on each side (the 2 st loops) the same way you work all the other stitches. For the last stitch (two st loops) at the end each row you slip the first one purl wise with the yarn in back, and the last one slip purl wise with the yarn in front. That is it for edges. It makes a very nice edge with one side one color and one side the other, no mixing of colors.
I also used Fallingblox’s 2 color cast on, but it is not necessary. I wanted to add a row of white at the bottom and top, but because I didn’t heed something Fallingblox explained about his cast on, I ended up doing 3 rows at the beginning (and will put 3 at the end). But I think 2 would be nicer and match the 2 stitch side border best.
when it says to slip the first stitch does that mean to slip it knitwise with the yarn in the back?
If you do yours the way I did mine you won’t need to do this. I don’t really know which way they want you to slip it.
Good luck on your first DK project. After my first try I swore I’d never do it again, but I did and it has gotten easier. At first it seemed so slow and confusing for a method of making something as mundane as a potholder. I thought it would have been a lot easier to make 2 and sew them together. :lol: But then I wouldn’t have learned to DK. 
It’s a fascinating technique.