EDIT: Due to the apparent confusion, I've rewritten my answer.
This is a job for the transfer principle! The basic idea is that every standard object and notion of standard analysis can be translated perfectly to non-standard model.
In particular, the standard notion of limit applies equally to the non-standard model: simply by applying the *-transfer to the ordinary notion of limit, we get an operator
$ \lim_{x \to a} f(x) $
where $f$ is an (internal) function and $a$ is a hyperreal, and the dummy variable $x$ ranges over hyperreals near $a$.
Again by the transfer principle, this operator even agrees with the usual $\epsilon$-$\delta$ definition (with everything varying over hyperreals rather than reals). Although this fact is less useful: one usually doesn't use $\epsilon$-$\delta$ arguments to prove things about non-standard limits of non-standard functions: one instead proves things about standard limits of standard functions, and then transfers those results to the non-standard model.
Of course, this only applies to internal statements. External ones don't transfer: e.g. the property that
$ \lim_{x \to 0} f(x) = \text{st}(f(\epsilon)) $
for nonzero infinitesimal $\epsilon$ can only be expected to hold when $f(x)$ is a standard function.
The incompleteness of the hyperreals is not a problem here, because that is an external fact. The hyperreals, in fact, satisfy the internal version of the completeness axiom -- e.g. every bounded (internal) subset of the hyperreal numbers has a least upper bound. You just have to be careful to work with internal objects. e.g. $\mathbb{N}$ and $\mathbb{R}$ are external subsets of ${}^\star \mathbb{R}$, so it's not surprising that they don't have least upper bounds. In fact, this relates to the overspill principle -- e.g. any subset of ${}^\star \mathbb{R}$ that contains all of $\mathbb{R}$ must also contain an infinite hyperreal.
Limits at infinity or converging to infinity, such as
$ \lim_{x \to +\infty} x^2 = +\infty$
also transfer. As usual, one can work either with the ad-hoc definitions of such limits given by introductory classes, or one can transfer the extended real numbers. The latter yields an interesting conceptual fact: the infinite hyperreals are not "beyond $+\infty$": the standard extended real number $+\infty$ is still larger than every hyperreal number. They infintie hyperreals merely closer to $+\infty$ than any standard number, in a sense very similar to the fact that infintiesimals are closer to $0$ than any nonzero standard number.
As for your first question, you are correct. In the representation of the value of ${}^\star g(y)$ as a sequence of real numbers, each component is indeed a standard limit: in particular, the standard limit that defines $g(y_i)$.