I really like how the speaker really says that, they don't just speak it. It's one of the reasons I appreciate the course has an actual speaker and not text-to-speech, as limited as that makes it, it makes it more real and the tone of the voice helps me remember a lot of words and phrases.
Sometimes cad is used with an explicit copula (cad is ainm duit?, cad is brí leis sin?), sometimes it's used with another verb (cad tá uait?, cad a thárla?), and sometimes with a preposition (cad chuige?), sometimes with an implied copula (cad é an t-am é?)
In this case, the copula is explicit, and cad is ainm duit? is just how you cay it. An alternative version, cén t-ainm atá ort? relies on an implicit copula (cén is a contraction of cad é an).
You can't say "cad bhfuil mo ainm?" for 2 reasons. The first is that you would use the independent form of the verb, tá, not the dependant form bhfuil (cad tá ar siúl?). The second is that "name" and "Paul"(for example) are both nouns, so you need to use the copula is, not the verb bí.
The most common alternative to Cad is ainm dom? is Cén t-ainn atá orm? (which uses bí rather than the copula because of the preposition ar).