EDIT (original answer below):
WHuber's comment is dead on. A stochastic process is a statistical description of the values. A realization is a particular set of values that adhere to this description. You can create an ensemble of randomly generated realizations for a given process. (In physics we talk about ensembles. Do probabilists?)
In the ensemble, ALL values in the process are (or at least can be) random variables. That being said, you could speak of the subset of all realizations such that the first day is sunny. Perhaps this is a new stochastic process based on the original, but conditioned on the first day being sunny. I think this is what I hastily put in my original answer.
I suppose one could define a discrete stochastic process such that every 10th value is equal to $\pi$ or something, in which case these would have a fixed value in each realization. Such a process would not strictly fit the definition in your question. But that seems somewhat pathological.
No. The state of the first day is non-probabilistic. If whether it is sunny on subsequent days depends on the first day, then the probabilities are:
$$P[S, 1] = 1$$
P[S, 2] & = P[S, 2 | S, 1] \cdot P[S, 1] \\
& = P[S, 2 | S, 1]
P[R, 3] & = P[R, 3 | S, 2; S, 1] \\
& = P[R, 3 | S, 2] \cdot P[S, 2 | S, 1] \cdot P[S, 1] \\
& = P[R, 3 | S, 2] \cdot P[S, 2 | S, 1]