You can store the protein at 4C for less than 24 hours only. If you need to store for more than one day to weeks, use -20C and -80C for longer period of time.
Enzymes survive better in buffers that have a certain pH. I don't know your optimum buffer conditions. Straight water is not advisable. To look at auto catalysis there would be two methods. The enzyme has to be pure. First look at degradation over time with SDS PAGE gel . Second, do an enzyme activity assay to see if your protein is loosing activity. If it is a crude preparation, then any pro teases present can degrade your enzyme. It is best to add a protease inhibitor cocktail mix. You can buy pills, premade by many companies. But the key is to first find conditions where your enzyme is stable. So you will need to empirically test many buffers, etc if the work is not published yet.
To answer your basic first question, I think the way to study auto proteolysis is to make a mutant protein that does not have catalytic activity with a tag, and mix it with wild type enzyme. If processing occurs in trans, the mutant protein will get processed. If the mutant protein sitting by itself does not get processed, then it is auto catalytic. You can have both things occurring in trans and auto catalysis, but you would know this by these experiments.