I am in process of getting my Green Card. I am having approved asylee status. Can I travel to Schengen States with this travel document and can I return back to the US safely?
The entry ban to the US of people who have been to the Schengen Area in the preceding 14 days is still in effect. Until that ban is lifted, if you travel to the Schengen Area, you will not be able to return to the US unless you first go to a country other than the Schengen Area (and other than the UK or Ireland, Mainland China, Iran, and Brazil) for 14 days. The proclamation does not exempt people who hold US Refugee Travel Documents from the entry ban. The only exemptions are if you are certain types of immediate relatives of a US citizen or permanent resident.
In addition, I believe the Schengen Area is currently banning entry of non-EU and non-Schengen-Area nationals, so I don't think you will be able to travel there in the first place.