If the tap water in your area is safe, it should make no difference if you drink it from the kitchen sink or bathroom sink. However, if you are concerened about the safety of you tap water in general, contact your local water utility.
No. The water from your bathroom sink is the same water that comes from your kitchen sink and your shower and even your toilets. It is all safe to drink unless there are extenuating circumstances that require you to filter your water before drinking it. Some refrigerators will do this for you.
There is nothing wrong with that as long as your faucet is clean and the water in your home is drinkable. Sometimes bathroom faucets get build up from the minerals in the water and can cause the water to taste bad.