I am a 5'7" 158 pound female. I'm looking to lose 10 pounds. Is a 1600 calorie diet good for that?

Maybe. Losing weight is a matter of using up more calories than you take in. Exercise is a major component and should be balanced by modulating your diet. Eat 3 small meals, give up all snacks and sugar-containing soft drinks, walk or run several miles a day, and engage in toning exercises and you will lose weight.
It depends. It depends on how active you are. As a ballpark number, most people with average activity need about 10 calories (actually kcals) per day per pound of body weight to maintain their weight. More if they are more active. So, with average activity, 1600 calories would keep your 158 lbs stable. I would increase exercise and/or decrease calories by another 300-500 kcals / day.