Translation:She works in the supermarket because she needs money.
It's not. Teastaigh can be translated as "need" or "want" (though I would tend to use it more with "need", but I'm not a native), but the structure involving ó can only be used to express "want". Please report it.
Not to dissent. In the NEID, under 'need", the first example given is Tá 10€ uaim'. The boundary between need and want can be tricky and in some cases synonymous. "I need to hear an apology from you,' etc
Looking more carefully at the examples, i think you're right. When it's clearly need they are talking about, like the chikdren needing sleep, it's teastáil ó. When it's ambiguous, it can be either, apparently.
Airgid is the genitive of airgead. You will see it used in the phrase a chuid airgid, meaning "his (share/portion of) money" or bonn airgid meaning "silver coin" (or "coin (made) of silver"). Bonn can also mean "medal".
The genitive is also used after the verbal noun, so ag tuilleamh airgid is "earning money".