1. I think you should choose a weight lifting program. Lifting heavy weights will help build muscle (and lose fat at the same time in some cases) and make your body composition more appealing. There are some good beginner's programs out there, like the New Rules of Lifting for Women.
2. Weight lifting is resistance training, and I think it would be the best thing for your situation. I've lost a fair amount of weight as well and it's the weight training that has given me a nicer body composition at my lower weight.
3. Usually 3 times a week is fine
It depends on the program. New Rules of Lifting for Women is 3 days a week.
4. Only if you aren't getting in enough protein from other foods. It is debated a bit how important it is to have that protein right after working out, but I like to do it myself. I do drink the shakes because I can't always get my protein from food and it's easier.
5. Whatever protein shake that fits into your calories and macros! I use a few different kinds, one that is 20 g of protein and 85 calories and another that is 30 g for 130 calories. I mix them in unsweetened almond milk because I like the taste.