Yes, you are correct. If someone else edits a file that you have open, a second one will be created (with the date and the name of the computer next to it). I'm not sure how to lock a file from others - since you're using drop box as a group, but you could simply work out a schedule with the Prop Master. Perhaps to say that the Props Master cannot edit the file during rehearsal hours, and to be aware that during rehearsal, the file is changing, so it's best to look at it before or after rehearsal (and make their updates then). That's the best I can think of. The other option is during rehearsal, remove the file from Drop Box, and when rehearsal is over, put it back.
I love using Drop Box for shows. Generally I prefer to work on my iMac at home, but when I'm in rehearsal, I need to use my laptop. With the show saved on DropBox I can do both, and not have to worry about USB flash drives or emailing things to myself, and I never get confused about which version is the most recent. I find it's super helpful with my ballet work, as I have DropBox on my work computer - and I can access it from my laptop when I'm in the theater and from my iMac when I'm at home. That has saved me tons of emailing files to work on over the weekend, major file transfers over the network, etc. AND...I have DropBox on my iPhone, so if someone needs a certain piece of paperwork while in rehearsal, I can pull it up on my phone instead of going back to my office.
I also use it in my personal life, as I have a folder shared with my husband and we keep things in there that both of us could use at any time. We work as "co-webmasters" for a restaurant's website, so I keep all files related to the restaurant in that folder, so either of us can update the website from anywhere.
I am indeed a DropBox addict....