Independent stochastic processes and independent random vectors

The answer to all your questions is yes. And they can be deduced from the following :

If two random vectors $\boldsymbol{X}:=(X_1, \ldots,X_n)$ and $\boldsymbol{Y}:=(Y_1, \ldots,Y_m)$ are independent, any pair of "marginalized" random vectors $\boldsymbol{X_A}$, $\boldsymbol{Y_B}$ (each formed by arbitrary subsets of the originals) are independent.

This property (basically your second question) can be readily deduced from the definition of independence (factorization of joint densities) and marginalization. From this the other two follow.

Take $t_1, t_2, \ldots t_n, s_1,s_2, \ldots s_m \in T$ and define $X := (X(t_1), \ldots, X(s_m)), Y=(Y(t_1), \ldots , Y(s_m))$ . Now by definition $X$ and $Y$ are independent.Then take two projections $\pi_1 $ and $\pi_2$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^{n}$ and $\mathbb{R}^{m} $respectively such that $\pi_1 \circ X =(X(t_1), \ldots, X(t_n))$ and $\pi_2 \circ Y =(Y(s_1), \ldots, Y(s_m)).$ Now as projections are continuous, both $(1)$ and $(2 )$ follows.