Most procedures show that the enzyme should be kept in the buffer solution and the pH should be according the stability of the enzyme. I dilute the enzyme in distilled water and it needs to be kept at fridge temperature (4 degree Celsius) for the stability. Please refer to the product specification regarding the storage condition of the enzymes. Most enzymes have to be kept at lower temperature.

Yes you can dilute it with water and yes you can store it at RT if you want to promote a hydrophobic effect, which will produce a protein precipitate. This aggregation is reduced by using low temperatures.