Random Process at a particular time instance
Your equation (3) and (4) are wrong.

When you say 


  Also, the sum of probabilities of a random variable should be 1


you actually mean the total sum of the values of probability mass function $P_{X_k}[x]$ equals one; i,e.

$$ \sum_{k} P_X[x_k] = 1 $$

(or the integral of the probability density funtion $f_{X_k}(x)$ equals one as
$ \int_{-\infty}^{\infty} f_X(x) dx = 1 $ )

But in your equation (3) and (4), you are not summing the values of probability mass function but the values of the sample function itself, which leads to te wrong conclusion and confussion. 

To reach the correct conclusion, in your example RP, the pdf associated with each RV $X_{t_k}$ is 
$$
f_{X_k}(x) = P(H)  \delta( x- \sin( \omega_1 t_k) ) + P(T) \delta(x- \sin( \omega_2 t_k)) 
$$ 

which integrastes (sums) to one as

$$
\begin{align}
\int f_X(x) dx &amp;= \int \left( P(H)  \delta(x- \sin( \omega_1 t_k)) + P(T) \delta(z- \sin( \omega_2 t_k)) \right) dx \\
&amp;= P(H) \int \delta(x- \sin( \omega_1 t_k)) dx + P(T) \int  \delta(x- \sin( \omega_2 t_k)) dx \\
&amp;= P(H)+P(T) = 1
\end{align}
$$