For the knit, if you're going to use any stabilizer, I'd consider Pellon 30 Lightweight Stabilizer (though I'm not really sure you need a stabilizer unless either the knit is too thin to the point is might pull apart/become thread bare, or if you plan to quilt the knit fabric).
For the sweater, I definitely would not use a stabilizer. It will make a very uncomfortable pillow having stabilizer on both sides. I've made pillows with knits & ones made entirely from a sweater & have never needed stabilizer. The only thing you do need to consider is that if you are not buying a pillow form, you will want to create one using cheesecloth or the lightest weight muslin you can find. The pillow form will keep the fill from getting caught in the seams or oozing out through any gaps in the sweater's knitting. If the sweater is loosely knit, you will want to make sure to either dye your fabric to match, or pick a fabric color for the pillow form that will coordinate (if it's a dark color, you can go with dark grey or black; if it's a light color, tan or cream will usually work). I use Mountain Mist Fiber Fill for my pillows & it creates a nice fluffy pillow. They do make cotton fill; I just haven't tried it so I can't comment on it. I've made pillows out of all sorts of fabrics & have never done anything with the fabrics before sewing them together except wash, press & cut them.
The only thing I would suggest is to make sure you use a Ballpoint needle in your machine. I always use Gutermann 100% polyester thread when I make my pillows as it comes in lots of colors & gives me a nice strong seam. I use Aurifil cotton for quilting, but pillow seams get more stress on them so I worry about their stability. I set my stitch length to 15+ stitches per inch (1.7mm or less). Fair warning: if you ever need to pick out that seam, it will be unpleasant but the seam will be very strong.