I do not crop. It is part of the way I see and work in the landscape. It is a personal choice, but I do not hold it as a religious belief that all should follow. And I do not think less of someone's cropped image than I would of someone's full-frame image. I do some cropping in-camera by means of a modified darkslide that gives me two 4x10 images on an 8x10 piece of film. And "one of these days" I will modify a darkslide to make 8x8 images on 8x10 film. I do love the square.
I do not dodge or burn either, again from personal choice. It has nothing to do with "purity"...it is just the way I see and work with the light.
Jim J. -- I am deeply concerned about getting the best possible print, which includes the image. In fact that is my photographic goal -- to be able to see intensely enough to find the best possible image that does not require cropping, burning or dodging. I contact print using either carbon printing or platinum/palladium printing. And for the record, when I was silver gelatin printing (the first 15 years of my 35 years of photography), I would sometimes crop and did a lot of burning and a little bit of dodging, so I am quite familiar with the techniques. In fact, I can remember doing 25 sec base exposures, and then spend the next 10 to 15 minutes burning in the image. I saw the paper as a piece of marble and I would use the light to chisle down into it to revel the darker insides.
For beginner photographers, shooting and printing full-frame is a good exercise for training one's eye...for quickly learning to see as the camera/lens sees. For the same reason, using only one prime lens is also a good exercise for beginners. Both help them to be aware of what is happening in the corners and sides of the image -- those are important as they define what is happening in the center.
PS...below is a heavily burned in image on the left (Truman Cove, South Island, NZ). And a "straight" pt/pd print on the right (El Capitan Meadow, Yosemite Valley). Both are full-frame.