Item 1 is true. Problem 22 on page 217 of Horn and Johnson [HJ94] states that for any unitarily invariant norm ||⋅|| and a positive semidefinite block matrix $\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}$, it holds that $\left\|\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}\right\| \le \|A\|+\|C\|$. Instantiate this with the Ky Fan norm to get the desired inequality.
(I do not have the access to the book right now, and I used the paper by Audenaert [Aud06] to get this reference. [Aud06] contains a proof of a generalization of this inequality.)
Item 2 is false by considering the case where A=C=I/2, B=0, and k=1, where I is the identity matrix. (Did you mean to square the left-hand side?)
References
[Aud06] Koenraad M. R. Audenaert. A norm compression inequality for block partitioned positive semidefinite matrices. Linear Algebra and its Applications, 413(1):155–176, Feb. 2006. http://dx.doi.org/10.1016/j.laa.2005.08.017
[HJ94] Roger A. Horn, Charles R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1994.