Can a periodontist check the margin where a crown meets a tooth root to check for decay that has caused a fistula? Or, they do not do this--only pull?

Can examine. . The periodontist main concern is the condition of the gums and supporting structures(bone, etc.). However most do thorough examine as well as hygienist who is cleaning your teeth that look for most problems. The fistula is a bump or pimple on gum that is drainage vent for an infection. It's source is usually an abcess under tooth. Root canal may be needed not filling or new crown. .
Any dentist can. Fistula is a sign that there is an infection and your body is draining it out. Unlike some other types of infection, a dental chronic abscess will not get better on its own. See your dentist-periodontist for diagnosis and treatment.
Periodontist. A Periodontist is a gum/bone specialist. The Specialist can check crown margine, but this a decision better made by your Restorative Dentist. Fistula may be from infected root canal, infected gums, or both. You may need root canal Rx or extraction. Then attention must be paid to replacing any missing tooth material. See your General Dentist to coordinate treatment.
Decay and Fistula. If the decay on the margin of a crown caused a fistula to form due to an abscess, it is easily diagnosed by any dentist or specialist. An abscess would require root canal therapy and a new crown. Periodontist do not only extract teeth... on the contrary, they are educated and trained to save them. There comes a point, however, that teeth may break down and become hopeless.