I'm confused. Dropbox does file-sharing, not remote desktop. Also, your requirements seem to want your computers to not be synced, hoever Dropbox syncs across all clients - so differentiating between computers (e.g. "a list of all available computers") is not necessary.

Unfortunately I do not know of any out of the box solutions. You could setup samba shares on each computer and if you want to access them anywhere then proxy through Amazon. Otheriwse, I'm hoping this might help you better phrase your request.