This is done in multiples of three. I would say size 9 or 10 needles. The bottom part is knit one, purl 2 for I'd say 8 rows or two inches. The top of the hat is similar to a pair of baby pants I just finished today. It was done in a thermal rib. Mine was done in a multiple of four. Yours would be (assuming you're doing this in the round) knit 1, p2 for two rows. Knit two rows. Repeat these four rows for pattern. If done on straight needles, knit 1, p2 for the first row. Second row: k2,p1 across. Row 3: Knit across. Row 4: Purl across. Seam it in the back when finished. Hat height will be about 10 1/2 inches if you want this to cover your ears. Rather than do decreases, you'll probably be better off knitting this straight until the proper height is reached, gathering up all the stitches and drawing tight, then fastening off. It's going to look really cute. I believe Lion brand has a natural shade of Fisherman's Wool that is similar to that one.
Welsh corgis are cool.