If I grind Crushed Red Pepper how does the result compare to, say,Cayenne, Chili Powder, White Pepper, Cayjun, etc.? I know they all are very similar, is there any reason that one shouldn't grind crushed red pepper and use one of the other options? Is it just a matter of personal taste?
First of all, white pepper is not similar to red pepper at all. White pepper and black pepper come from peppercorns, not peppers. I'm not sure what you mean by comparing "Cayjun" to crushed red pepper. If you mean Cajun as in the ethnic cuisine found in Louisiana, that's a style of cooking and is not just one spice, or even an exact recipe of spices. Cajun seasoning could include a number of spices such as the ones listed here: http://en.wikipedia.org/wiki/Cajun_cuisine#Seasonings
Chili powder is another example of a seasoning blend with different kinds of spices. You can see an example blend here: http://en.wikipedia.org/wiki/Chilli_powder
Crushed red pepper (or red pepper flakes) is simply dried flecks of red chili peppers, seeds and flesh. From the items you listed, the only two that are interchangeable are crushed red pepper and cayenne since they are both the same species of plant, Capsicum annuum, and are singular spices rather than blends. If you grind crushed red pepper, you will end up with cayenne pepper for the most part.
What is the reasoning to use crushed red pepper over the other medium heat flavoring?
Could you give an example of a medium heat flavoring? The are a number of reasons why one might use a different seasoning that just happens to also be less spicy (hot), but it's not always about the heat. Paprika for instance has a fruitier flavor than cayenne in addition to being less hot. In some case, like what I think you were alluding to, it's easier to handle one spice versus another. There is also the benefit of accuracy when using a spice with less heat. If you want to add a very small amount of heat, it's easier to add it through a milder spice because the measuring doesn't have to be as precise.
If you are infusing an oil with crushed red pepper, I see no real benefit to grinding it first. Keep infused oils in the refrigerator. I don't use infused oils, so I've never witnessed the lifespan of one. It will probably have a faint musty odor when it's time to discard it. I definitely wouldn't keep it around any longer than I would the seasonings if they were left on their own. I would also check for rancid or stale smells from oil oxidation, and taste it for off-flavors before using it after you've had it around for a month or two.