Q1 and Q2 have a positive answer for all groups (I assume $G$ is a countable group). Let $\mu$ be a proper function from $G$ to the positive reals, and view it as a discrete measure on $G$. Assume in addition that $\mu$ grows reasonably, and more precisely satisfies an equality of the form $\mu(gh)\le C_g\mu(h)$ with $C_G\>0$ (e.g. fix a proper subadditive length $|\cdot|$ and define $\mu(g)=|g|+1)$.

Then the action of left action of $G$ on itself induces a well-defined left regular representation $\pi$ of $G$ on $\ell^2(G,\mu)$, which is bounded ($\|\pi(g)\|\le C_g^{1/2}$).

Let $e$ be the unit in $G$ and $\delta_g$ the Dirac function at $g\in G$. Define $b$ as the coboundary $b(g)=\delta_e-g\delta_e=\delta_e-\delta_g$. Then $\|b(g)\|\ge \mu(g)-\|\delta_e\|$, which is proper; thus $b$ is a proper cocycle.

Q1 and Q2 have a positive answer for all countable groups (conversely a discrete uncountable group cannot bear any proper function). Let $\mu$ be a proper function from $G$ to the positive reals, and view it as a discrete measure on $G$. Assume in addition that $\mu$ grows reasonably, and more precisely satisfies an equality of the form $\mu(gh)\le C_g\mu(h)$ with $C_G\>0$ (e.g. fix a proper subadditive length $|\cdot|$ and define $\mu(g)=|g|+1)$.

Then the action of left action of $G$ on itself induces a well-defined left regular representation $\pi$ of $G$ on $\ell^2(G,\mu)$, which is bounded ($\|\pi(g)\|\le C_g^{1/2}$).

Let $e$ be the unit in $G$ and $\delta_g$ the Dirac function at $g\in G$. Define $b$ as the coboundary $b(g)=\delta_e-g\delta_e=\delta_e-\delta_g$. Then $\|b(g)\|\ge \mu(g)-\|\delta_e\|$, which is proper; thus $b$ is a proper cocycle.