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The second claim is false for $n=3$. Choose $\varepsilon$ small and $\delta\ll\varepsilon$. Let $A$ be the set of all points $(x,y,z)\in\mathbb R^3$ satisfying the following inequalities: $$\begin{cases} -1.5+\varepsilon+\delta &\le x+y+z &\le 1.5+\varepsilon \\ -1.5+\delta &\le x+y-z &\le 1.5 \\ -1.5+\delta &\le x-y+z &\le 1.5 \\ -1.5+\delta &\le -x+y+z &\le 1.5 \\ \end{cases}$$ The integer translates of this set cover the space, but its $\ell_1$-diameter is no greater than $3-\delta$.
Added. The first claim is false too. In the above example, fix $\delta=\varepsilon/10$ and add the inequality $$\max\{|x|,|y|,|z|\}\le 0.5+10\varepsilon$$ to the system.
The second claim is false for $n=3$. Choose $\varepsilon$ small and $\delta\ll\varepsilon$. Let $A$ be the set of all points $(x,y,z)\in\mathbb R^3$ satisfying the following inequalities: $$\begin{cases} -1.5+\varepsilon+\delta &\le x+y+z &\le 1.5+\varepsilon \\ -1.5+\delta &\le x+y-z &\le 1.5 \\ -1.5+\delta &\le x-y+z &\le 1.5 \\ -1.5+\delta &\le -x+y+z &\le 1.5 \\ \end{cases}$$ The integer translates of this set cover the space, but its $\ell_1$-diameter is no greater than $3-\delta$.