"I earn money when I work with my father."
Translation:Tuillim airgead nuair a oibrím le m'athair.
agus followed by a pronoun is used in Irish in places that English uses "while", but when agus is a conjunction between two otherwise complete phrases (as it your suggested answer - Tuillim airgead and oibrím le m'athair) it would not mean "while".
Tuillim airgead agus mé ag obair le m'athair could be interpreted that way, but it doesn't mean exactly the same thing as the sentence in this exercise.
In almost all cases, a follows nuair (the exceptions are when the phrase is negative, like nuair nach éisteann tú - "when you don't listen", or when it is followed by a version of the copula - nuair is am dúinn imeacht - "when it is time for us to go").
When the verb after the a is tá, they are joined, to give atá, but it's still the same relative particle a.
Leis would be used with a following article, e.g. … nuair a oibrím leis an athair (“… when I work with the father”). Other possible combined forms are lena, e.g. lena athair (“with his father”) and lena hathair (“with her father”), and lenár, e.g. lenár n-athair (“with our father”).
airgid is the genitive form of airgead.
There are a number of different ways to decide when you need to use the genitive in Irish - it is usually involved if you are saying "of money" in English (tá a lán airgid aige - "he has a lot of money"), but it is also used when a noun immediately follows another noun in Irish, so when you encounter cuid airgid, roinnt airgid, beár airgid, nótaí airgid or lucht an airgid, you will know to use the genitive airgid because it's the 2nd of two nouns. (lán is also a noun).