Are there different types of vertigo?

Absolutely. Vertigo is a sensation of movement when you are not moving. It is a symptom, not a disease. The length of the spinning spells helps us figure out what the problem is. Spells lasting less than a minute could be bppv. Spells lasting less than 24 hours may be migraine or meniere's disease. Spells lasting longer may be a viral inner ear infection. See your ENT specialist to figure this out.
Yes and no. Vertigo will always be vertigo (false sense of motion or false sense of room motion). But, vertigo can be caused by many different things. In general, you can have either peripheral (inner ear problem) or central vertigo (brain/brainstem problem). Then out of each of these, there are many causes of vertigo. Ents and/or neurologists are the best doctors to see for an evaluation of vertigo.