Need some clarification!

I am going to start knitting some Kitty Slippers and it calls for the following “Inc 1 st in first st. K1”. Does this mean to knit the first stitch, leave on the left needle and then knit again through the back loop? and then of course, knit the next stitch???

Yes, it does-- you’ve got it. But I prefer the more hidden increase and use it in most cases, no matter what the pattern calls for. On this site, the video is called M1F-- you lift the bar of yarn between the 2 sts onto your needle and knit into it in a way which twists the bar, so that the increase is pretty much invisible.

Yes, that’s right. If you’re on the first row and there’s no yarn between the sts yet, except the CO, then do the kfb or the M1A which is the backward loop inc.