I looked at the Slim ROM zip and in system/app I can see DownloadProvider.apk but not DownloadProviderUI.apk.
E: anybody else who doesn´t have Downloads-'app'? Very annoying to downloand something when browser crashes 'cause I don´t have that.
I looked at the Stock+ ROM zip and it has both.
Judging by this link the first provides the Downloads functionality while the latter provides the UI for the user to interact with it:
Perhaps you could download the Stock+ ROM zip, extract the DownloadProvideUI.apk from it and then install it on Slim ROM to see if the app icon then appears?
Obviously do a Nandroid backup in CWM/TWRP before doing anything like this so that you can revert to your working setup.
Hope this helps.
Edited by dalyer, 05 August 2013 - 10:35 AM.