Can the iOS Dropbox app download all contents of all folders to my iPad? Is it possible to force Dropbox on iPad to sync all files?

(That is, to make it working it the same way as it does on a laptop/computer.)

As far as I see, it saves only files marked as "Favorite". Adding everything manually does not seem convenient...
Not really. It isn't designed to store everything, but does try to cache files you have downloaded and save them as long as the iOS device doesn't run low on space and tell apps to throw away files that can be re-downloaded. 

Keeping more than a Gigabyte of free space on the filesystem will keep iOS from purging the files dropbox has downloaded.