Mmmm.. Camera care. This is one of those things will vary from owner to owner and is directly proportional to the degree of OCD exhibited by the owner.
I like to use a very soft 1-2" paint brush (which has never been used for anything else) to clean camera bodies and such.
Lots of people use canned air to blow dust off lenses and viewfinders. Personally I've never figured out how to use those effectively without bursts of frozen moisture going everywhere. I have an older style bulb blower that is sufficient for clearing the dust from a lens before actually cleaning it. Always use air or a lens brush (love my lens pen, but there are a couple of DPs for whom I work who I won't use it near. Their way or the highway.

) to get the chunks off your glass before hitting it with a cloth.
Lessie. Lots of case options out there, and depending on your location, hard cases/flight cases/etc that are a good size for a DVX are probably easy and cheap to find second hand.
I would never leave a tape in overnight. That's a personal preference thing. It's probably fine to do so from a technical standpoint (no clue honestly), but logistically it sounds like a nightmare waiting to happen. What if you were watching footage on it at the end of the day, and wound back to the middle of that day's shooting. Then in a rush the next morning just started rolling without checking to see if the tape was queued to the right spot.
Suddenly you're overwriting yesterday's work. Sure, you could try to keep track, but better to take it out, make sure it is labeled, hit the copy protection and put it away for safe keeping.
Similar for batteries, personal preference. I like to take them off at the end of the day, but not everyone is a stickler for this. I also like my batteries numbered and dated (date new), but I'm pretty OCD/anal.