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No. Take the Hilbert space $H={\ell }^2$ and its standard orthonormal sequence $(e_n)$. Letthe unit sphere. This set is norm closed but is not convex. By the Bessel inequality, for every $y=(y_j)\in {\ell }^2$,so $e_n\rightharpoonup 0$; equivalently, this is a weakly null orthonormal sequence. Every $e_n$ lies in $C$, but the weak limit $0$ does not. Hence convexity is essential.

No. In $H={\ell }^2$, let $x_n=e_n$ be the standard orthonormal sequence. For any $y=(y_j)\in {\ell }^2$,so $e_n\rightharpoonup 0$. However,for $n\ne m$. No subsequence is Cauchy, hence none converges strongly. This is a weakly null orthonormal sequence.