Is it fair for me to drink caffeine during tests? For some of my more difficult and early morning tests, I will buy coffee and bring it with me into the exam. My hope is that the caffeine will help me be fully alert, think more clearly, and write answers more quickly.

However, I know that drinking caffeine puts me in a slightly different state of mind physiologically than my classmates. Is it fair for me to drink caffeinated beverages at tests? If it isn't, is it merely something rude that would be nice for me to stop, or a form of mild cheating that I would be wise to admit to and discuss with my professors?
That would be totally acceptable. Remember, you are also in a totally different state if you went to the toilet right before the exam. If you drank a litre of water in the morning. If you ate something for breakfast that happens to contain lots of short-chained carbohydrates, providing lots of energy for your brain.

You have to draw the line somewere and coffee is definitely acceptable, if not expected.

Edit:
As several comments suggest, your other question about rudeness because of smelling beverages: that's different from course to course, and you will probably have to employ some degree of common sense there. People are usually not bothered by coffee smell as opposed to french fries or the sounds of eating potato crisps. If in doubt, just ask your TA or fellow students before going to the exam.