I don’t think the key is correct in referring to a single or double cast on. The “single cast on” looks like a left slanting decrease (purl 2 tog through the back loop). The “double cast on” is a double decrease, likely a centered double decrease.
There isn’t evidence in the chart for increases at those symbols. Rather they are decreases that balance the yarn over increases so the stitch count remains the same.
The rows between the pattern rows may be knit row to maintain the reverse stockinette of the pattern rows but I can’t tell for sure because of the resolution of the photo. You could work a swatch of the chart to see what looks good to you.
For repeats of the chart, there are going to be 8 repeats along the row. When you finish rows 1-11 (maybe the pattern means through row 12 since you probably need the return row 12) of chart 1 repeat the rows 1-12 again. before going on to rows 13 and 14.
This is somewhat of a guess since the pattern leaves things out and perhaps loosely translates others. Definitely swatch.