Characterisation of norm convergence
Solution 1:
for Q2 - in general it holds for uniformly convex spaces
Solution 2:
For Q1: Here's an example in the Banach space $c_0$ of sequences of (real or complex) numbers converging to $0$ with sup norm. Let $x_n=(1,0,0,\ldots,0,0,1,0,0,\ldots)$, with $1$s in the first and $n^\text{th}$ spots and $0$s elsewhere. Then $(x_n)$ converges weakly to $x=(1,0,0,\ldots)$, and $\|x_n\|=\|x\|=1$ for all $n$, but $x_n$ does not converge in norm.