The Sun is the name of our star just like Sirius and Alpha Centauri are names of other stars.
We don't call teams 'the Collingwoods' or 'the Sydneys'.
Yes we do, eg "we'd like our team to a play a brand of disciplined team footy like the Sydneys or the St Kildas of the competition."
Perfectly grammatical. Making a semantically singular word plural is generally fine. Making an uncountable noun like "milk" or "rice" is not.
Irrelevant though because "sun" is a noun that can be semantically and grammatically plural. Look it up in a dictionary.
It's capitalised because it is the name of a team, not because it refers to a plural of our Sun.
If it was the Gold Coast Fishes, then maybe you would have a point.