I realise the third person singular is used for groups in the absence of a pronoun (Mae Alys a Jon yn Cymraeg) but why is this 'ydy' and not 'ydyn' ??
It's only ydyn if the verb comes before the pronoun:
Y pencampwyr ydyn nhw "They're the champions"
Nhw ydy'r pencampwyr "They're the champions"
Likewise as mizinamo mentioned from above:
Yr athro ydw i "I'm the teacher"
Fi ydy'r athro "I'm the teacher"
In this kind of construction, it's ydy for all persons -- as you can see from shwmae's comment above, it would even be e.g. Fi ydy'r athro for "I am the teacher" (It's me who's the teacher), rather than, say Fi ydw'r athro.
Nage, ni ydy'r pencampwyr.
The answer for sentences or questions that don't start with a verb is either ie "yes" or nage "no".
That's correct. They're used with the simple past and are the only two options. Answering in the past is nice because unlike with other tenses you don't have to specify a person.
Why not "maen nhw" for they are? Just when i think im getting Welsh it confuses me again!
Haha, yes it is confusing!
You're right that most sentences in Welsh meaning "they are" will start with maen nhw, for example:
Maen nhw'n bencampwyr "They are champions"
Maen nhw'n athrawon "They are teachers"
Maen nhw'n blant "They are children"
Notice in the above sentences, what "they are" is always something indefinite - "champions, teachers, children". If you want to say that "they are" something definite - "the champions, the teachers, the children" then Welsh doesn't allow you to use maen nhw. Instead you put nhw "they" at the front and ydy "is/are" after, giving you:
Nhw ydy'r pencampwyr "They are the champions"
Nhw ydy'r athrawon "They are the teachers"
Nhw ydy'r plant "They are the children"
You can't say something like Maen nhw'n y pencampwyr to mean "They are the champions". You have to use a structure like Nhw ydy'r pencampwyr instead because y pencampwyr "the champions" is definite.
Nope. If you want to say "They are" + a definite noun, you can't use the Maen nhw construction. It has to be either Nhw ydy'r pencampwyr "They are the champions" or Y pencampwyr ydyn nhw "They are the champions".
The same applies to other persons: Fi ydy'r athro or Yr athro ydw i, not *Dw i'n yr athro; Hi ydy fy ffrind or Fy ffrind ydy hi, not *Mae hi'n fy ffrind.
Are we allowed to put the subject of the sentence before the verb in all sentences?
Technically you can, but the subject only comes first for particular emphasis. Ordinarily the verb begins a normal non-emphatic sentence.
One pencampwr, several pencampwyr.
One of many nouns (usually [always?] referring to people) where the singular ends in -wr and the plural in -wyr.
Another pair is dysgwr, dysgwyr "learner(s)".