I don't know what most of those ailments are, but apparently they were quite Common in Colonial New England, because Dr. Calcott traveled through the region for the next four years, advertising his "Art of Curing" in all the local newspapers.
Although there were plenty of local physicians in the area in the 1700s, Dr. Calcott was part of a wave of itinerant doctors and healers who appeared in New England at this time. Most were men and came from Europe, although one Native American woman traveled through Essex County claiming to cure cancer through herbal treatments. Like Dr. Calcott, most also advertised through newspaper ads, but some relied just on local word of mouth.
These traveling healers were denounced by the more established doctors and often advocated highly unorthodox treatments. In addition to herbal treatments and elixirs, electrical devices were frequently used, often in front of a paying audience. These electrical demonstrations were part medicine and part entertainment.
Dr. Calcott didn't perform in front of an audience, but he did use a particularly strange technique. Here is an account of how he cured six year old Elizur Belden of blindness:
"Mrs. Belden holding the Child in his Lap, Dr. Calcott ... Licked the eyes, first putting his Tongue into one Eye and then into the other Eye of the Child - it was soon done, - and instantly the Child saw, and ever after continued to see well."
That account was written by Ezra Stiles, the President of Yale, after talking with Elizur Belden's parents many years after the cure. The testimony is reliable. The licking seemed to be effective, but why?
The modern scientific mindset would probably chalk it all up to psychosomatic illness, but Isaac Calcott claimed he had special healing powers because he was the seventh son of a seventh son. In many parts of Europe it was (and still is) believed that the seventh son of a seventh son has special abilities. Often these were healing powers, but in some places seventh sons could also predict the future and find lost objects. Wikipedia has a short but interesting article on this belief. In contemporary America, the seventh son motif is still popular in genre fiction.
The licking was also probably considered effective because spit was known to be a fluid highly charged with personal power. The average person's spit was often used in simple folk magic, but as the seventh son of a seventh son Dr. Calcott's saliva was obviously quite powerful. Jesus spit in a blind man's ear to make him hear; Dr. Calcott licked a blind boy's eyes to make him see.
Unfortunately the old adage "Physician, heal thyself" seems to apply to Dr. Calcott. Despite, or perhaps because of, his amazing powers Calcott was an alcoholic. Once, in Middletown, Connecticut, the good doctor was so drunk that two young boys were able to tar and feather him. He was last seen practicing his medical magic in 1773 in Portsmouth, New Hampshire before disappearing from the historical record.
I found this information in Peter Benes's article "Itinerant Physicians, Healers, and Surgeon-Dentists" in Medicine and Healing. Volume 15 of the Proceedings of the Dublin Seminar for New England Folklife.