Maybe you ought to seriously crop the absurd amount of academic questions you pose on these forums. :p
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.
I usually print a full frame version though so I can live with it for a bit before I decide on the final cropping. Make some cropping squares (L shaped pieces of paper) and try out many crops. I think each image will have a strongest crop, and it usually isn't what the camera dictated.
I never crop personal works. I crop for clients that want an 8x10 out of a 6x6 neg. I try to convince them to just get a 8x8 instead though.
Occasionally, I'll be out with the 4x5, but come across an image that should be square, so I'll crop in the darkroom. My basic rule is to try to use one entire dimension (i.e. the full 4 inch, or 5 inch length) of the negative. I want to utilize as much real estate available on the negative, but the final image is what's important. For those that don't crop, I'd ask how many shots you've passed up because you don't have the lens/format/point of view required for a particular image?