Foot usually means the one by the needle. (Foot pedal would be the other one.)
A "walking foot" or "even-feed foot" is a special foot you buy for the machine. The "feed dogs" are the moving teeth that move fabric through the machine. A walking foot feeds the fabric from above too, which makes both layers of fabric move together more evenly. A walking foot is very helpful for certain sewing tasks.
A twin needle is used mostly for decorative top-stitching. It is not used much for quilting. It is used mostly to stitch parallel straight lines on clothing hems.
It is better to buy a good used sewing machine than a cheap new one. If the one that came up in the picture is your machine, I think you may become frustrated with it.