Since the surreal numbers form a saturated real-closed field, it follows that they serve as a (proper class) version of the hyperreal numbers, including the transfer principle. This means that absolutely every function and relation on the real numbers extends to the surreal field, with furthermore all the same first-order expressible properties. (This is provable using the global choice principle in Gödel-Bernays set theory or Kelley-Morse set theory.)
$$\langle\mathbb{R},+,\cdot,0,1,<,\mathbb{Z},\sin,\exp,\ldots\rangle\prec\langle\text{No},+,\cdot,0,1,<,\mathbb{Z}^*,\sin^*,\exp^*,\ldots\rangle$$
So we will have a surreal sine function, cosine, exponential, logarithm, and so forth—every function whatsoever has a surreal analogue with the same first-order expressible features in $\text{No}$ that the original function has in $\mathbb{R}$.
In particular, yes, there will be a periodic nonstandard function $\cos^*$, with all the same properties of the $\cos$ function on the reals—it will be periodic with period $2\pi$ and obey the expected identities with other trigonometric functions, $\sin^{2}(x)+\cos^{2}(x)=1$.
The nonstandard analogues $\sin^*$, $\exp^*$, however, are definitely not unique except in trivial cases. One can see this immediately because the surreal field has a superabundance of automorphisms, and there is nothing special about $\omega$, for example, in the field structure alone, for it can be moved by field automorphisms to any other infinite element. In light of this, there will be many distinct versions of the nonstandard functions, with all the same expressible properties. In this sense, it is not directly sensible to ask, what is the value of $\cos^*(\omega)$? In fact there are versions of $\cos^*$ for which this is $0$, or $1$, or $-1/\sqrt{2}$, and so forth, simply because of the automorphisms.
Therefore, when asking for surreal versions of the various standard functions, one should specify more exactly what is desired. In particular, how do you want the function to interact with the genetic birthday structure? The birthday structure is not determined by the algebraic surreal structure. The surreals are not saturated with respect to the birthday structure, and the field is rigid when one incorporates that structure as a part of what is meant by the surreals.
For this, there may be some good accounts of which $\cos^*$ to use, and I shall leave this for others.
Finally, you ask about the power series, but in the surreals there are issues with this. First of all, as I recently explained on another answer, the surreals do not have any nontrivial convergent sequences or series at all, when considered in the ordinary meaning of sequence or series, with countably many terms. A countable sum of nonzero terms never converges in the surreals. Indeed, even uncountable series sums of set size never converge—every set of surreal numbers is discrete in the order.
However, there will be nonstandard analogues of the power series, with $$\cos^*(x)=\sum_{n\in\mathbb{N}^*}(-1)^n\frac{x^{2n}}{(2n)!},$$
with terms proceeding into the nonstandard even natural numbers. The trouble here, however, is that $\mathbb{N}^*$ is a proper class in the surreals — each instance of this series is a proper class — and so there are some delicate set-theoretic issues in handling them. Many things work, but one must take care. (Meanwhile, again, the interpretation of $\mathbb{N}^*$ itself is not unique.)
