Proofs with strong approximation theorem

For your first question about a rational $\hat{\Bbb{Z}}$-basis of $gL$.

Take $m\in \Bbb{Z}-0$ such that $mg$ is in $M_n(\hat{\Bbb{Z}})$. Let $h = Adj(mg)\in M_n(\hat{\Bbb{Z}})$.

Then $gh = cI$ with $c=m^{n-1}\det(g)\in (\Bbb{A}^{(\infty)})^\times$.

So $gL$ contains $c L$.

$cL=c'L$ with $c'\in \Bbb{Q}^\times$.

$c'L$ has finite index in $gL$ and the $J$ elements of $gL/c'L$ have representatives in $\Bbb{Q}^n$.

Therefore $$gL = c'L + \sum_{j=1}^J v_j \hat{\Bbb{Z}} = (\sum_{i=1}^n e_i c' \Bbb{Z}+\sum_{j=1}^J v_j \Bbb{Z})\otimes_\Bbb{Z} \hat{\Bbb{Z}} = (\sum_{i=1}^n w_i \Bbb{Z})\otimes_\Bbb{Z} \hat{\Bbb{Z}}=\sum_{i=1}^n w_i \hat{\Bbb{Z}}$$