My bottom teeth started moving and it hurts. Should I see an orthodontist at 50?

Absolutely. Seek an initial examination (usually free) with a qualified Orthodontic Specialist. The Orthodontist can then team with your General Dentist and, if advisable, a Periodontist to create a multidisciplinary treatment plan to resolve your problems. A full one third of all Orthodontic patients are adults. Take advantage of an Orthodontist's expertise.
Age for braces. Yes. I recently finished treating a 90 year old for the same reason.
Sure. You are going to have your smile for at least 30 more years. We have patients as old as 72! nO HARM SEEING AN ORTHODONTIST AND SEEING WHAT THEY CAN DO.
Possible at any age. Teeth that are crowding can be a cosmetic and functional problem at any age. Older patients (although 50 is not old) become more at risk for periodontal problems due to the difficulty maintaining adequate oral hygiene. Check with your dentist whether you might be a candidate for braces or invisalign. You should have healthy gums and adequate bone support around the teeth to be moved.
Sure. Orthodontic treatment is appropriate at any age. But first see your general dentist for an evaluation.
Maybe. First, you should see your dentist. You might have gum disease which is allowing your teeth to move or it could be occlusal trauma. Depending on why your teeth are moving there should be a game plan from start to finish and your regular dentist is best suited to be the manager of the game plan.
Maybe. Teeth do shift with age- but...If you are experiencing pain in the area you should see your dentist first for an evaluation and x-ray to see what is going on first. Tooth movement can be a sign of a pathologic process that may not be just innocent shifting.
Yes. Changes in bite and wear can cause a crowding and discomfort. Please, seek the care of a dentist that can address your concerns, some other conditions may have caused the situation; and then in turns the dentist will refer you to the orthodontist after these conditions have been cleared or controlled. Both oral health professional will work together to address your needs.