Your first assumption is correct. C standard has the lowest string at C1 and the rest of the strings tuned up a perfect 4th from each other. Drop C means that you've got it tuned to D standard, but leaving the lowest string at a C so it's a 5th below the next string rather than a 4th. You're only "dropping" the tuning of that last string. (On a guitar, sometimes you'll come across double drop tunings where the bottom string and the top string are both dropped - so double drop D for instance would be DADGBD. Neil Young used this on some of his songs.)
I find I play differently on my BEAD bass than I do on a 5 string BEADG bass. Unless you want the extra upper range of the G string - which most stoner/doomy stuff doesn't need - I think you'd be better off setting up one or more of your basses to C standard. You may need to get the nut slots filed a little wider to accommodate heavier strings. Get a luthier to do this if you're not confident in your own ability, you'll probably spend less on a single shop visit than on a full set of nut files anyway.
D'Addario makes a balanced tension set that goes from .050 to .120, which is probably right about where you want your gauges. You could go heavier if you prefer more tension, in which case any lighter gauge 5 string set will work - just don't use the top string. I wouldn't go over .125 for the lowest string.
I don't recommend using BEAD or BEADG tuning, you'll find yourself fretting that first fret on the low string a lot and that's going to tire you out quickly.