Hi mate,
Firstly 003 should be fine to use. Infact many use up to 010 to 015 before changing resin. Your technique is probably more important for your finishing result rather than whether your using 000 or 010.
Secondly, yes storing water in containers is fine. That is what those with a static tank at home do basically.
Thirdly, you ought to investigate why you are getting 003 as opposed to 000. Is this a new setup? How old is the resin?