This is a really good point. When you see a northern and a southern word, often one of them is also considered more "standard", e.g. southern standard llaeth (milk), gyda (with "company"), bachgen (boy), nawr (now) as opposed to northern llefrith, efo, hogyn, rŵan; and northern standard oeddwn (I was), mai (that), gan (with "possession"), i mi (to/for me) as opposed to southern o'n, taw, gyda, i fi. So it's good to be aware of both.
A lot of the standard forms are actually northern, or originate in the north. This is because when the Bible was translated into Welsh in 1588 it became widely-read and the basis for the modern literary language, and the translator, William Morgan, was from Tŷ Mawr Wybrnant in the north. I sometimes wonder what modern Welsh would be like now if he'd been a southerner.
Yep, it's the Romans: pons, ecclesia, fenestra.
There are many others e.g. lleng < legio, ystafell < stabellum; bresych < brassica, gwin < vinum, days of the week: dydd Sul, dydd Llun, dydd Mawrth etc. < dies Solis, dies Lunae, dies Martis etc. and some months: Ionawr, Chwefror, Mawrth etc. < Ianarius, Februarius, Martis etc.
The compliment marker yn is not used with eisiau (want) or angen (need). This is because they are not verbs - they're nouns! They are described, when used like this, as defective verbs, meaning they are used as verbs in colloquial Welsh but, grammatically, they are nouns.
The "proper" way to use eisiau and angen is: Mae eisiau X arnaf i where X is the thing you want, which can be a verb or a noun; Mae eisiau mynd arnaf i 'I want to go', mae eisiau afal aranaf i 'I want an apple'. This construction literally translates as 'there is a want for X on me'.
In the south they use an actual verb, mofyn (usually seen and heard as moyn) which uses yn: Dwi'n mofyn mynd 'I want to go'.
There are about four or five main dialect areas, not two, and many more local variations. The geographical and age boundaries between the dialects are very fuzzy and have large overlaps. There isn't a single dialect which covers all of 'north Wales', nor 'south Wales'. In the later section on 'Dialects', the supporting notes have some links to external sites which have more detail about the main geographical dialect areas. This is an example
This course does not cover any particular dialect in any detail. In the Welsh taught to adult beginners there are really not very many differences to be concerned about - the main thing is to become familiar with the basic structure of the language and some basic vocabulary. The structure of the language is what many beginners have some initial problems with as it is so different from English.
Accents do vary quite a bit, but you will quickly adapt to the local accent wherever you go.
When you do move, get in touch with the local Dysgu Cymraeg providers and start learning with them. The tutors will cover local dialect features as well as the more generic Welsh covered in course books.