In the southern course what is the beginning of the sentence which in the northern course goes: ai eich plant chi ydy rheina? Is it y fe, or a fe? And what does it mean? What is it a contraction of?
In the southern course again, you use the pattern: oes gyda chi blant eich hunan? I sometimes say: oes plant gyda eich hunan? Is that equally correct?
Sorry if someone has already asked these questions. 
Sounds like an āiā sound to me; so āi fe eich plant chi yw rheina?ā (may not be spelled like that). The direct translation would probably be close to āYour children, are those?ā
For the second question, Iād personally use oes plant gyda chi eich hunan - it seems like you use āyou yourselfā and āhe himselfā and so on as a pattern. Hence āDylai ti deimloān falch ohonot ti dy hunanā (You should feel proud of yourself).
I havenāt listened to it recently, but I would assume it is āifeā, which is an alternative for āaiā (I certainly have heard both round here in the South.) It doesnāt actually mean anything - or at least, has no equivalent in English. Like āaiā, it is simply used to mark the fact that this type of sentence is a question.
Also worth noting that, as well as both being acceptable, none is too. It can be, and is left out by some speakers.
So, like: eich plant chi yw rheina?
I recall questions used to have an a at the beginning, in place of the affirmative makers, mi/fe, and that negatives had na, or was it in? Incidentally I was also told while in Wales that the affirmative marker is the first person, mi, in the north and the third person, fe, in the south. I did therefore wonder whether ai/ ife was following a similar pattern. Though Cat says ife and Iestyn says what sounds to me like yfe.
I would say it is more that āaiā is regarded as the more āstandardā word wherever you are, and āifeā can be used as an alternative in the South (if, as robbruce says, any word is used at all!) rather than being a North South 100% shibboleth. Like many of these things!
I found this video on Facebook. It has two questions in the middle that begin with āaiā. I thought it might be useful to see, even if it doesnāt really answer the question in the op.
That video has an error. Ydw is I am, not I do. But, it kind of sounds like I do. Maybe thatās why they said that?
I think it is a matter of the difficulties of translation, rather than an error as such. They say quite clearly at the beginning that it is a form of the verb āto beā.
The thing is, in English it sounds very odd to say āare you understandingā (to which the answer is āI amā) , rather than ādo you understandā (to which the answer is I do). Using āare you understandingā as a translation of the question would have been odd, and thus using āI amā as a translation would have been odd itself. I understand the point you are making though!
Very good point. It could have been better explained though considering itās catering to beginners.