EDIT: A question on eigenvalues of non-negative matrices.
Solution 1:
This is an answer to an older version of the question:
Regarding one part of your question: Consider $A= \begin{pmatrix} 1 & 1\\ 1&1\end{pmatrix}$ and $B=\begin{pmatrix} 1 & 0\\ 0&1\end{pmatrix}$. These are satisfying your conditions but the smallest eigenvalue of $A$ is $0$ while all eigenvalues of $B$ are $1$.
Edit: $\det(A)=0$ and $\det(B)=1$, so this is also a counterexample for the other question.
One can easily check that both matrices are positively semi-definite, as required.