Yes it is perfectly normal
Your puppy is still very infintile. Usually they don't leave their mommas until 12 or more weeks old. He is missing his littermates and will take him a while to settle into this whole new world of his. Imagine a baby being taken from everything they know (puppy leaves the comfort of momma and littermates) and thrown into a big scary new world with strangers. It takes some time.
That being said, if you have him in a crate next to your bed, the best way to make him feel at ease is to have him UP at your level (on a nightstand, dresser, etc) so he can see you at night. Sometimes just reaching in to stroke him with your hand is all it takes to settle him down. You will still be getting up for potty breaks for some time though.
The rough estimate sometimes used is a puppy should be able to hold their bladder as many hours as months old they are. IE, if he is 3 months old, approx 3 hours between potty breaks. This is just a general rule of thumb, food, exercise, water intake are all variables in this formula
Hope this helps some, and congratulations on your new puppy!