About a domain of random variable $S_n=X_1+X_2+...+X_n$

In principle, one chooses a probability space: a set $\Omega$ (whose members are the individual "outcomes"), a $\sigma$-algebra $\Sigma$ of subsets of $\Omega$ (the events), and a probability measure $P$ on $\Sigma$. All the random variables you're interested in then correspond to $\Sigma$-measurable functions on $\Omega$.

In this case $H$ and $T$ label the possible outcomes for each individual coin-toss, but they don't capture the outcomes of the whole sequence of coin-tosses: heads on the first toss is not the same thing as heads on the second toss. Don't be misled by the fact that the same labels are used! To specify an outcome of the sequence of coin-tosses, you need to say which of $\{H, T\}$ occurred on each toss. Thus you take the set of $n$-tuples $\{H,T\}^n$.