I often see this confusion about LE, AI and Elements. They share the same installer but are different Versions. Wich one of those three do you have? If i am recall correctly the Groove Agent 4 SE is only available in Elements and not in AI or LE
So its AI 8 then. I quess GA is not part of AI 8. You can upgrade to Elements 8 for a reasonable price. It gives you Groove Agent and you can download the Accoutic Agent for Groove Agent as well.
This now depends. if you want to use your own samples than yes. Halion Sonic SE should come with some drumkits as well but you cannot use your own samples.
But you allways have the option to use third party VST(i) to fill this gap. there are plenty of free plugins and also commercial ones. What sounds actually are you looking for ?
When you open Halion Sonic SE and try to load a program you may be not seeing the filters. In the bottom left of the preset picker window there is the Set up Window Layout button. If you click on this and enable the filters checkbox you should be able to filter the preset type i.e. Drums&Perc
I think in all versions of Halion Sonic SE there should be some drum sounds although your choices might be limited in the lightest versions.