A Rotating Chord in a Tetrahedron

In a regular tetrahedron $D\text{-}ABC$ of edge length $1$, let $O$ be the center of $\triangle ABC$. A line through $O$ meets segments $AC$ and $BC$ at $M$ and $N$ (endpoints allowed); $P$ is the midpoint of $DM$. Determine, with proof, which of the following are true:

1. for some position of the line, $NP \perp$ plane $DAC$; 2. the maximum area of $\triangle DMN$ is $\dfrac{\sqrt{2}}{4}$; 3. the minimum of $\tan^2\angle DMN + \tan^2\angle DNM$ is $12$; 4. the ratio of the volumes of the pyramids $D\text{-}MNC$ and $D\text{-}MNBA$ ranges over $\left[\dfrac{4}{5}, 1\right]$.