2) If there were no proc coefficients, you'd have to make procs boring and useless.
3) Proc coefficients are necessary to ensure that your random low-cost, AoE spell does not outshine any other spell. That's why proc coefficients are constantly being tuned (e.g., WW in D3V for the CM wizard, Rain of Toads for the WD to proc Solanium, and so on). Tuning is an ongoing process and nowhere near perfect though.
4) I completely agree that proc information should be available in-game as part of the tooltip, and so should many many other in-game mechanics that are just poorly worded or have hidden mechanics such as internal cooldowns. Note that there is a way to retrieve the current proc coefficients - looking at this list, though it's difficult and tedious to get all the info out of it, and will soon be outdated as it's for 2.3. For most spells, the old 2.0.6 list is still viable though.