Let $V$ be a model of set theory with CH. Recall the following definitions.
A forcing $\mathbb{P}\in V$ has the Laver property if for any $\mathbb{P}$-generic filter $G$ over $V$, functions $f\in\omega^\omega\cap V$ and $g\in\omega^\omega\cap V[G]$ such that $g\le_*f$, there exists a function $G\in\big([\omega]^{<\omega}\big)^\omega\cap V$ such that $g(n)\in G(n)$ and $|G(n)|\le n+1$ for every $n\in\omega$.
A forcing $\mathbb{P}\in V$ preserves reals non-meager if $\mathbb{R}\cap V$ is a non-meager subset of $\mathbb{R}\cap V[G]$ for any $\mathbb{P}$-generic filter $G$ over $V$.
Typical examples of proper forcings having the Laver property and preserving reals non-meager are Sacks and side-by-side products of Sacks, Miller and Silver (-like). The properties are also preserved by countable support iterations.
Question 1: What are other (natural) examples of proper forcings with the Laver property and preserving reals non-meager?
Obviously, if $\mathbb{P}$ preserves reals non-meager, then the left half of the Cichoń's diagram must be $\omega_1$ in the extension. However, e.g., in the countable support iteration of length $\omega_2$ of the Miller forcing one gets that $\mathfrak{d}=\omega_2=\mathfrak{c}$ and of a Silver-like forcing that $\mathfrak{r}=\omega_2=\mathfrak{c}$.
Question 2: What other standard cardinal characteristics of the continuum can we make big using a proper forcing having the Laver property and preserving reals non-meager?
E.g. what about $\mathfrak{a}$, $\mbox{cof}(\mathcal{M})$ or $\mbox{cof}(\mathcal{N})$ (while $\mathfrak{d}=\omega_1$) etc.?
