Thanks for the ideas Dan! After a lot of trials, I finally found that the following procedure works:
1. Download Terrestrial Mammals and Marine Mammals separately (Thanks for this idea Dan, I hadn't thought of it for some reason)
2. Repair Geometry (as I had done before)
3. Select by location (so this selected only the polygons that were located in the area that I wanted to clip it to)
4. Export selected data into a new shapefile
5. Split by Attribute Plug In (Split By Attribute Tool - Desktop Decision Support Tools) which split each polygon in the shapefile into a different feature class of a new geodatabase
6. Clip using Model Builder's feature class iterator
And so far, it has been working!