Under any definition of normal that's useful, no, it isn't normal.
Mild inflammation for a day or two, yeah. Intense pain along with joint mobility issues, hell no. That's abnormal, and even if it turns out to just be a weird but short term response to the tattoo, it needs to be checked out by a doctor.
I'm not saying it never happens, because it does. You aren't the first person to experience it, but it isn't a standard, expected response to a tattoo. You expect topical pain, some swelling at and around the site of the tattoo, maybe some degree of joint issues for a day or two at the outside when it's on or very near a joint.
But a week later, with "crazy" pain and the joint not moving right is atypical for sure, and if the worst case scenario were involved, you'd want it checked out sooner rather than later
Infections can present like that, as can nerve damage. That's two things you don't fuck with. Both are treatable, but untreated can cause worse things to happen. Probably It's not a big deal, just an unusually intense response that's going to go away after another week at the outside. But it's my opinion that the risks of waiting and seeing are higher than the costs of a doctor visit.