Mm, dog (ci) is a subject, wants (eisiau) is a main verb. Mae is a form of to be. What exactly is its role in such type of sentences? Just a copula?
Well "Eisiau" actually isn't exactly a verb but a noun which we happen to use to express "To want". Though to answer your question, unlike in English, a form of "To be" is needed in this sort of sentence. It more accurately translates as "The dog is wanting to eat an apple" though this construction is not used in English so instead we translate it into "The dog wants an apple".
But if it was "Mae'r ci yn mynd i'r swiddfa" it would mean "The dog is going to the office", correct? Then mae (and also dw and dych and others) are copulas. But is it really a marker of present continuous here? Or just present tense?
Represents both the present continuous and present tense. E.g "Mae'r ci yn bwyta" can mean both "The dog eats" and "The dog is eating".
A better way to start to understand this pattern is to add a missing element:
- *Mae'r ci ag eisiau bwyta afal" - literally 'The dog is with (has) a want of eating an apple'
The ag is not seen except occasionally in a more formal register of the language.
In the informal register of the language there is no distinction between the present and present continuous. In the formal registers the tense system is different in some respects.