You don't want to disable cookies, you need to make sure that they're enabled. By default they are, so if you haven't changed any of the settings it should be OK. However, it's possible that Opera is blocking scripts (javascript) that HotFile and RS need in order to allow the file transfer. I can't give you any other information, but if you tried the download links in another browser (FF, IE, etc) and it works, then you know there is some incorrect setting in Opera.
Again, I would suggest trying JDownloader since it will automate downloading from all filehosts.