Show that $B(t)$ is a Brownian motion

Question

Let $W(t)$ be a Brownian motion, and define $$B(t) = \int_0^t sign(W(s))dW(s) $$ where $$ sign(x) = \begin{cases} 1, & \text{if $x \geq 0$ } \\ -1 & \text{otherwise} \end{cases}$$

1. Show that $B(t)$ is a Brownian motion?
2. Show that $E(B_tX_t)=0$

For question 1, I know how to prove $B(t)$ has independent increments, but how to prove its increments are normal?

For question 2, using Ito's product rule, $$ d(B_tX_t) = B_tdX_t+X_tdB_t+dB_tdX_t $$ Since $B_t$ and $X_t$ are correlated, how do I deal with $dB_tdX_t$ in above equation?

Thanks in advance.