>What's a sugar baby?
Someone financially supported by a sugar daddy/mommy in exchange for a sexual relationship.
>What's a trophy wife?
A much younger woman married for looks usually after divorcing one's first wife.
>Are these terms sexist?
Probably not helpful to label the terms sexist (given that they describe real phenomena) but the mentality that seeks out those relationships might be.
>If I date someone with student loans then am I a sugar daddy?
If you cover/displace significantly more of their finances than in a typical dating relationship without reciprocation, yes.
>Some of my friends have wives who stay at home but don't have children. How do you define those relationships?
Childless marriages, distinguished from sugar daddy relationship by the expectation of permanence and more complex mutual obligations.
>Do all cultures fetishize the young? SeekingArrangements is in UK and Singapore too.
Young women, yes, for fundamental biological reasons.