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While (as Keith suggests) you can use the Smith normal form, you can also use the Hermite normal form. Find (using integer row operations) a generator matrix for $\Lambda$ which is upper triangular. If the diagonal entries are $d_1,\dots,d_n$ then coset reps are the $\sum a_i e_i$ where $0\le a_i < | d_i|$.
If you have smooth compactly supported functions $f_i$ on a locally finite collection of open sets then $\sum a_i f_i$ converges to a smooth function for any $a_i$. :-) To get nice locally finite covers we need to exploit the circle of ideas around paracompactness/partitions of unity.
Petya, I'm sure you're correct, but it's a bit more fiddly to ensure the uniform convergence of the sum of the derivatives of the bump functions without a convenient global coordinate system.
Isotropy of $u$ and $w$ means that $q(u)=q(w)=0$ where $q$ is the quadratic form in question. In your posting you actually discuss the case where both are nonzero. For the counting argument, think about how many pairs $(x,y)$ of elements in $\mathbb{F}_q$ satsify $xy=0$? $xy=1$? etc.
