Apparently it's also related to the Italian grasso, the Spanish graso and the French gras (as in Mardi Gras. Give a cheer to Superjey.) All of which look back at the old Latin crassus (which means "thick, fat, overweight) (remember that next time you find some Roman guy named Crassius in a story or movie.)
Where it comes from before that? I have no clue.
As for a relation to grease? Quite possibly. but it's late and I need to get up in the morning.
According to etymonline.com: Yes. English "grease" derives from the same source as grasso/graso/gras.
Discrete piece vs mass quantity aside, you can tell that fat is the thing she doesn't eat because it's marked as a noun (-o) and as a direct object (-n).
Yes, adverbs end with -e, although not everything makes sense as an adverb. I can parse "She eats greasily", (and apparently "greasily" is a recognized word) but it's not really a usual thing to say.