Is it possible to have $m$ rows with $n$ variables over F2, where $m < n$
that will have some combination of rows that can form a zero vector, but there will be no zero vectors in reduced-row echelon form?
Please provide an example of such a setup if this is indeed possible.
Having $m$ rows and $n$ columns is obviously possible for matrices over $\Bbb F_2$, no matter the $m$ and $n$.
Gaussian elimination is an algorithm that works for matrices over any field and it is generally formulated in a way that may be applied verbatim in that generality (even when it is presented only for the case $A\in\Bbb R^{m\times n}$). I don't quite understand your objection, but I presume it revolves around a confusion between the action of $\Bbb Z$ over $V$, according to which you may write $x+x+x+x=4x$, and the action of $\Bbb F_2$, which is the one of interest and which only has $\overline 0\cdot x$ and $\overline 1\cdot x$.