How do I make the sweater actually look knitted?

Those little Vs on the pattern are the whole answer. Real stockinette knitting is just rows of interlocking Vs, so if the sweater reads as a red blob with lines, it's because the lines aren't forming the V shape. Back stitch rows will never get you there, no matter how neat they look up close. What worked for me on a similar tiny animal in a sweater was two steps. First I filled the entire sweater area with satin stitch in the main color, working it as smoothly as I coud. Then I went back over that base with the Vs in a slightly diferent shade, working each V as a pair of tiny straight stitches angled toward each other, and offsetting each row by half a stitch so the Vs interlock like brickwork. Done in rows bottom to top, that gives you that cozy ribbed look from any distance. Keep the Vs small and consistent, like two or three millimeters tall, and don't worry about perfect symmetry, real knitting is a bit wonky anyway.

If you want to go fancier, Ceylon stitch is the proper embroidery stitch that actually looks like knitted fabric, but at two inches tall on a mouse I found it too fiddly and the straight stitch version above does the job fine. Stem stitch in tight rows gives you a nice twisted rope look but it still reads as cords, not knitting, so I'd skip it for this. On thread, two strands is right for the base fill on something that small. For the Vs I dropped down to one strand, which lets you get the detail in without turning the torso into a lumpy mess. And yes, your instinct is right in general, thicker thread for the clothes and thinner for the fur. For the fur itself, just short straight stitches radiating outward from the body work better than split stitch at that scale, since you want a soft fluffy edge rather than visible line work. Do the fur last so it overlaps the sweater edge and looks like he's actually wearing it.

One trick from my crewel days that hasn't let me down yet, work your little V rows over a gridded guide. Light hand with a water soluble marker, just faint horizontal lines where each row of Vs sits, and they come out even without you squinting yourself blind. Washes right off when you're done. I did a cardinal in a little scarf years back doing it that way and the texture held up fine at that size. Also, don't skip the blocking afterward, a light press face down on a towel fluffs the whole thing up and settles any wonky rows. Mouse sounds charming, hope you post him when he's done.

I tried that little V idea on a gnome ornament last month and my rows kept sliding sideways halfway down, like the whole sweater was leaning. Turns out I was going too big with them, once I made each one tiny it finally looked like actual wool instead of zigzag scribbles. Also tracing the shape onto the fabric first helped a lot, I could see where the sleeves needed to curve before I got lost in all the little marks. Do you work the rows from the bottom up or start at the collar and go down?

Update: tried the gridded guide with faint horizontal lines and the tiny Vs tip, and it finally reads as knitting instead of scribbles. Rows stopped sliding sideways too. Still stuck on thread weight though. Two strands feels chunky on a two inch mouse. Do you keep the sweater and the fur the same thickness, or thin the fur down?