Conditional expectation of this stochastic process?
When you define the $W_n$ stochastic process you're not just defining it on the space $\{H,T\}$, because that space would only contain information on one coin toss. You define it on the space of all possible outcomes of all coin tosses. Your space is $\Omega = \{\omega = (\omega_1, \omega_2, \dots) | \;\; \omega_i \in \{H, T\}\}$. Therefore $\sigma(W_1, \dots, W_n)$ is not the entire $\sigma$-algebra.