I've been doing all of these proofs the same basically, I just want to make sure I'm doing them right, I didn't include all the details but I have the outlines of my proofs here.
1) U and W are subspace of V s.t. V = $U \oplus W$. Prove $u_1, \dots, u_m, w_1, \dots, w_m$ is a basis of V where the u's are a basis for U and the w's a basis for W.
I said that since the basis for U is lin ind in V then you can extend it to be a basis by adding the vectors from W. Since they are linearly independent as well, then you add every vector. This shows that $u_1, \dots, u_m, w_1, \dots, w_m$ is linearly independent. It obviously spans V because V = $U \oplus W$, there fore it is a basis
2) Prove or give counter example. $v_1, \dots, v_n$ and $v_1, \dots, v_n$ are linearly independent in V. Then:
$\lambda v_1, \dots, \lambda v_n$ with $\lambda \in F$ is linearly indp.
$\lambda (a_1 v_1 + \dots + a_n v_n) = 0$ (pulling out $\lambda$ from the span, we know the span is linearly indp). Therefore $\lambda 0 = 0$.
$v_1 + w_1, \dots, v_m + w_m$ is linearly indp.
Rearranging the span of this you get: $a_1 v_1 + \dots + a_m v_m + a_1 w_1 +\dots a_m + w_m = 0$ Split this into span(U) = $0$ and span(V) = $0$ implies all the coefficients must be zero.