Since I move to a new office I want to transfer all the content of my current backup server to Dropbox Pro and I am worried about loosing all the links in my Indesign files, as well as Autocad files etc.
The software you refer to usually save absolute path in links, as @Rich mentioned already. For software using relative links should not be any issue. So you shouldn't think about relative, but absolute paths in your links! This could be achieved placing your Dropbox folder on the same place (drive/partition and path) like the one used in old office, if possible. 😉 In such a way everything in Dropbox folder will keep the same absolute path like before.
I am a bit unaware of things, but what happens if I just copy all my folder directely to Dropbox without changing any names to any folders (except the "original" hosting folder which then would be Dropbox instead of our current Server...)?
See, doesn't matter how you would copy your folder to or from! The only important thing here is: are all paths referred from everyone project point to the valid place (direct or indirect) in the new office or not? You have to take care for the absolute path for every one component (according to the configuration used). That's it.
Add: Would be the same for moving using Dropbox, flash drive, DVD-r, or any other way.