From many sources I am hearing that symmetric polynomials not only an algebra, but also Hopf algebra. Would someone be so kind to explain where the coproduct (antipode) comes from ? And what it is useful for ?

It is mentioned e.g. here: Avatars of the ring of symmetric polynomials

Let me pay some attention to details of the question. Is it correct that there is some "natural" Hopf structure OR different authors writes about different and they are interesting each one in each specific task ?

Is it important to go to the limit n->infinity (which I heard many times, but not really like/understand) or we can stay in the more custom setup of $C[x_1,...,x_n]^{S_n}$ ?

What is puzzling for me that $C^n$ can be seen as an additive group and so $C[x_1...x_n]$ is Hopf algebra, but this does not seem to survive on $C[x_1...x_n]^{S_n}$. E.g. $\Delta(xy)=xy\otimes 1+x\otimes y +y\otimes x + 1\otimes xy$ which is not in $C[x,y]^{S_2} \otimes C[x,y]^{S_2}$.