should I wear tennis shoes/sneakers when walking in place on carpet or just socks?

Whichever makes you more comfortable. Some people cannot exercise without shoes. But if you like being in your sock feet, you don't need shoes -- especially on carpet.

I exercise in my socks at home.

You weren't born with shoes on. Of course shoes are good for outdoors but your feet need to breath like the rest of your skin on your body. And believe it or not try massaging the soles of your feet while watching TV. You'll be amazed how much better you feel in a short period.

Of course you can if you have a good pair of tennis shoes, here I'm giving you a link source having some good and colored tennis shoes.

If you are excercising, yes you should wear exercise shoes.