1) What does "confines" mean in "the gilded confines of the Beauty Hall were not my preferred habitat"?
"to confine" used as a verb means to be trapped or imprisoned.
In this sentence "the gilded confine of..." refers to be stuck somewhere that might be beautiful to others but it isn't for you.
2) What does "stuck out" mean in "She stuck out her bottom lip, nodded once, slowly"?
"to stick out" used as a verb means either to move forward from its usual place or be seen easily in its surrounding environment.
In this sentence is means that she moved her bottom lip forward in a way that indicates deep thought.
3) What does "obscenity" mean in "... a limp lettuce leaf and some salty tangy white slime which bordered on obscenity"?
"to be obscene" used as an adjective means to be disgusting or vulgar, usually in reference to rude language or nudity.
In this sentence, it means that the meal was really horrible and didn't even look like it was edible.
4) Do you say "supermarket trolley" or "Supermarket cart"?
American English is "cart" and British English is "trolley"