I recently found a very convincing and elegant proof of **Q1** in  *Balzin*'s **[Reedy model structures in families][2]**. In the paper, it appears as **Cor. 3.41.**

The *idea of the proof* is that it is enough to show that the infinity category associated to a model category $\mathcal M$ has $\infty$-colimits over directed Reedy diagrams, because they imply the existence of pushouts and coproducts. (Dually for limits). In order to do so  he uses that when $R$ is directed Reedy, $\mathcal M^R$ has always a model structure (without any additional assumption on $\mathcal M$) in order to build the Quillen adjunction, $$ \text{colim}: \mathcal M^R \leftrightarrows \mathcal M : c_R.$$ To finish he applies on of the main theorems of his machinary to derive the behaviour of the derived functors, **Thm. 3.37** in the paper. 







  [2]: https://arxiv.org/abs/1803.00681