1 - That depends on whether you were knitting with one color at a time. If you were knitting with one color at a time, then you would just bind off with just one color. If you were knitting with two colors at the same time, then continue to bind off with both colors.
2 - You can just do a simple garter scarf. That's what I did when I started color work. I used two different shades of pink.
:arrow: Cast on with dark pink, knit 2 rows. Add light pink and knit 2 rows. DO NOT CUT YARN, simply let go of the dark pink and just start knitting with the light pink. After 2 rows of light pink, pick up dark pink and k 2 rows. Keep switching until scarf is as long as you want it.
You can tie it on, but you don't have to. You can just start knitting with the new color. The first couple sts will tend to be loose, so you'll have to hold the new color's tail for a few sts. At first, you can hold that tail in your left hand with the left needle. But as your dexterity progresses, you can hold it in your right hand with the working yarn, if you knit English. If you knit conti, then you can hold in your right hand and progress to your left. But it really doesn't matter which hand you hold the tail as long as it's comfortable Some people even knit with both the tail and the working yarn for a couple sts. I personally don't like to do that, but try and see if you do