[250.56]. No. This is a commonly misunderstood point. The NEC doesn't require the earth resistance to be any particular number. The actual number is an engineering decision. The NEC simply says that if you drive one ground rod and the earth resistance is 25 ohms or less then you need to supplement it with an additional grounding electrode. It's common for electricians to pee on the rod and then measure--the salty urine is a strong electrolyte that brings the reading way down from just dry soil. If ever called upon to repeat the measurement, they pee first and then get people gathered around to witness the testing.
 

[250.58]. No. Doing so would create a dangerous difference of potential. In fact, they must be grounded to the same electrode(s).
 

[250.60]. No. Air terminals are of entirely different construction and dimensions. You can use grounding electrodes in lieu of air terminals, but you can't go the other way around.
 

[250.62]. No. Such pipe can serve as an auxiliary electrode and it should be bonded to your grounding system to eliminate dangerous differences of potential. But you cannot use it as part of the grounding path. It may seem wasteful to run a length of copper or aluminum wire along a water pipe instead of just hooking onto it on each end, but it's not wasteful at all. The pipe is for carrying water, not for carrying lighting or other undesirable electrical current.
 

[250.62]. The answer depends on what materials your cathodic protection system is constructed of. Use the appropriate conductor based on that, remembering that it must be "resistant to any corrosive condition existing at the installation." On any give site, the cathodic protection systems and grounding system must be engineered as a unified system, not as separate systems.