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    $\begingroup$ I think the following works: Let $\kappa$ be the least measurable cardinal in $V$. For 1), take $\lambda>\kappa$ such that $\lambda^\omega=\lambda$, then adding $\lambda$ many random reals will produce $2^\omega=\lambda$ and $\kappa<\lambda$ is real-valued meas. For 2), you can start with $V=L[U]$ and choose $\lambda=\kappa$ as in the first case. If there is a real-valued meas below $\mathfrak{c}=\kappa$ I think you produce an inner model of measurable cardinal $<\kappa$, this will contradicts uniqueness. For 3), any model of CH works. $\endgroup$
    – Jing Zhang
    Commented May 2, 2018 at 16:55