Yes. Anal tumors may arise form the skin and result in squamous cell carcinoma. Some of these tumors are caused by hpv infection. Tumors on the inside of the anal canal may arise form the gland, adenocarcinoma. There are additional infrequent tumor types as well.
Anal cancer. Yes different types of anal cancer are based on cell type seen under the microscope when biopsy is taken.
Most can. A rectal exam usually looks for trace blood (before stools looks red), which is usually from colon cancers. Can also check for internal hemorrhoids. Any primary care (gyn included) or surgical provider can do this test. Psych / opthalmology...Not so much.
Primary, gyn, gi. If you are concerned about a mass in the anus them your primary care doctor or gyn can evaluate to determine of this is a true rumor or hemorrhoids. Male patients are evaluated after 40-50yo for prostate nodules with a digital rectal exam. If suspected a gastroenterologist or colorectal surgeon will need to evaluate the mass for biopsy and pathologic evaluation.