For (i), I believe I am correct in saying that $|W_t|$ is a folded normal distribution with mean $\sqrt{\frac{2}{\pi}}t$ and variance $t$, and $tW_t$ is distributed normally with $\mathbb{E}(tW_t)=0, \text{Var}(tW_t)=t^3$?
For (ii) I'm not confident since I have only ever covered discrete-time martingales but using an analogous argument to the discrete case where we check if $\mathbb{E}(X_{n+1}|X_0,...,X_n)=X_n$, I am going to hazard a guess that the first one is a martingale, the second isn't?