First off, welcome to the forums...now to answer your questions...
1. The only benefit of using ice, is to cool the smoke. But you can have the same fantastic session with just chilled water. I stick my vase in the fridge about ten minutes before starting the session to chill the water.
2. NO! Do not stick it in the fridge...the best way to store it is at room temperature in an air-tight container or zip-lock bag. Store it away from sunlight too...
3. I think herbal shisha is crap. It doesn't provide the same flavor and enjoyment that you can get out of tobacco/shisha... The majority of the available shisha's out there have 0% tar and 0.05% nicotene...that's barely anything, so I would just stick to the shisha that's intended for hookah.
4. Cleaning wise, you're ok as long as you brush the inside of the stem and vase clean. If you leave any lingering smell, it can affect the next session in a negative way. Members around here like using a combination of baking soda and lemon juice to scrub out their hookahs. Also rubbing alcohol works wonders at eliminating any lingering smell.
I'm not sure how new to hookahs you are but check out the following links for some good info and advice on smoking hookah:
Also, make sure to thouroughly look through the review section. Most of your basic questions can be answered there...