The answer to my second question is "YES" NO". To show this let us use the following:
Lemma. Let $L$ be a lattice in $\mathbb R^q$ (and therefore $q$ is any positive integer). Assume $$\operatorname{diam} \mathbb R^q/L>1000.$$ Then there is a midpoint $m$ of two points in $L$ such that $|m-x|>1$ for any $x\in L$?
Modulo Lemma one can construct an action of parallel translations the answer following way:Let us construct inductively a sequence of lattices $L_q$ on $\mathbb R^q$ such that $\mathop{diam} \mathbb R^q/L_q<1000$ and such that $|x|>1$ for any $x\in L$.Start with standard $L_1=\mathbb Z$ in $\mathbb R$.To construct $L_{q}$ take $$L_{q}'=L_{q-1}\times \mathbb Z\subset \mathbb R^{q-1}\times\mathbb R = \mathbb R^{q}.$$If $\mathop{diam} \mathbb R^q/L'_q < 1000$ set $L_q = L'_q$.Othewise pass to the first on "NO").minimal lattice which contains $L'_q$ and the midpoint provided by the Lemma.Applying this construction finitely many times you get a lattice $L_q$ with $\mathop{diam} \mathbb R^q/L_q<1000$.
Continue the process, we get lattice $L_\infty$ in $H$ which is a $1000$-net, its fundamental doamin contains a ball of radius 1; i.e. $H/L_\infty$ is not compact.
Proof of Lemma.
