Let $ \begin{bmatrix} A& B \\\\ B^* &C \end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$.

Are the following plausible inequalities true? I have seen a lot of similar results, but for the following inequalities, I cannot locate them in the literature or find that they have been pointed out to be false.

1 $$\quad \sum\limits_{i=1}^k\lambda_i\begin{bmatrix} A& B \\\\ B^* &C \end{bmatrix}\le \sum\limits_{i=1}^k\left(\lambda_i(A)+\lambda_i(C)\right)\quad, $$ where $1\le k\le n$.

2 **Modified**
$$\quad \prod\limits_{i=1}^{2k}\lambda_i\begin{bmatrix}
A& B \\\\ B^* &C
\end{bmatrix}\le \prod\limits_{i=1}^k \lambda_i(A)\lambda_i(C) \quad, $$
where $1\le k\le n$.

3 $$ 2\lambda_i^{1/2}(B^*B)\le \lambda_i(A+C),$$ where $1\le k\le n$.

Here, $\lambda_i(\cdot)$ means the $i$th largest eigenvalue of $\cdot\quad$. Any references or counterexamples are appreciated.