I think that Gateau should be accepted as an English word. Here in England everyone knows what a Gateau is, it is a common word spoken here. It just refers to certain kind of cake rather than all cakes.
English has borrowed many words from french, particulairly in dealing with food, and subtly altering the meaning. Gateau for a type of cake, haricourt for a type of bean.
No difference. Context will tell you. I.e., "se gâteau ....", what does that mean? "Ce gâteau....", ah, now I get it!
There are two audible clues that will help you distinguish them. 1) the "hard c" vs. the "hard g", 2) the "d" vs. the "t". Try Google Translate: type in "cadeau" and press the speaker button to hear it pronounced. Listen very carefully. Now try "gâteau". After a few times, you will hear the difference clearly. Sometimes in audio exercises you will be able to pull out words that are less familiar simply by listening carefully.
I believe that is a mistake on the reader's side. You dont pronounce the s in the word est.
I'm just wondering how to differentiate between this or that. So far in these lessons I've only seen that ce/cette can mean either one. Or did I miss something?
No, you didn't miss anything. Both "ce" and "cette" can mean either this or that. Later on, there will be other lessons which will give you additional tools to be more specific when necessary.
On this side of the pond (U.S.), we call it a "cookie", on the other side (U.K.), it's called a "biscuit". That is, besides being the common word for "cake", of course. http://www.larousse.com/en/dictionaries/french-english/g%C3%A2teau/36216
"Ce gâteau est pour eu" - this was wrong, in a listening exercise. I intended a singular rather than plural cake recipient. It's there some sound difference between eu and eux I should have picked up on, or did I fail on a syntax level?