The veins, simply put, are the digestive system of the shrimp. You don't have to remove it, but if you don't want your shrimp to taste sandy or off putting it's best to remove it. you can use a toothpick to remove it quite easily.
Shrimp also works really well ground up and mixed with pork for a patty or a burger. It also does really well when battered and dipped into almost any aioli or flavored mayo. You can do a really good shrimp curry as well. Use tomato too, for a nice stir fry or a stew. Do a kebab and grill it up with a sauce of your liking whether bbq, hoisin, sweet chilli sauce, add it to your spinach and avocado for a great salad (poach or boil it), It also does really well in cream sauces.
For your last question, as soon as the shrimp turn pink it is fully cooked. They will get tough and rubbery when over cooked. Shrimp is a seafood and most seafoods cook super quick. Stir‑fry large shrimp for 3 to 4 minutes; medium 2 1/2 to 3 1/2 minutes, and small/bay 2 to 2 1/2 minutes. This is just a general guide though.
Best of luck.