Yes, you can copy object editor data (file -> export <type> settings) and triggers (right-click -> copy) from one map to another.
That being said, I wouldn't recommend having multiple people work on a map at the same time and copy stuff between their versions as it's likely to turn into a big mess that requires more work than it's actually worth. If you want to do something like this, a much better approach would be to have 1 person work on a certain aspect of the map (like terraining) and then send his version to another person to work on something else (like triggers).
The easiest way to do this would be for everyone to download Dropbox, put the map in there and then share it. This is not without problems: one person might mess something up and then you have no backups. Or you might work on the map at the same time, and then save over each other's work.
A more advanced (but also more safe) way of working would be to use w3x2lni and put your map on GitHub. Then you could work concurrently and have backups for every change ever made. I dunno how familiar you are with git but I can provide more details if you feel it would be worth the time investment.