Android - Dropbox app not installed, but there is log activity in /data/system/dropbox

/data/system/dropbox is not part of the cloud storage provider "Dropbox". It is part of the Android OS DropBoxManager (not related at all to "Dropbox", just happen to have the same name).

This puts data from application crashes and kernel logs and such in this log directory. I know the market app uses this directory (See #4) as well, probably to report application crashes.

You could probably remove the data in this directory, but do not delete the directory and you may see logcat errors from parts of the OS/Apps looking for specific files in this directory.

I deleted everything from this folder and rebooted my device afterward, I had no issues with any applications, and it actually freed up some space on the device.

As already answered, what you find in this folder are logs for events or crashes of the Android system. You could open those with a text editor and try to figure things out. All can be safely deleted, but the directory will get populated again as time goes by.

settings put global dropbox_max_files 9

If you are rooted (or via adb shell - thanks @Irfan Latif) you can issue the command above to limit dropbox files in that dir to 9 (or any other number you prefer). You will only see only latest n events, older get overwritten.

For me, 9 is a good compromise that let me see i.e. latest app crash event, without filling that dir with hundreds of small files. You will not run into any issues by setting that to 0, but you will not be able to see potential errors/crashes (silent or not) in apps that could be worth investigating and fixing.