I have read that while comparing atoms in a row we compare their electronegativities to know which atom would better stabilize a negative charge on it, and for atoms in a column we compare their sizes. Why is that so? Like, for example if we consider Nitrogen and Oxygen (which are same row elements), why can't we say "as Nitrogen atom is bigger in size, it can better stabilize a negative charge rather than the Oxygen atom" (as Nitrogen has more room for stabilizing the charge), but we say it is the Oxygen atom which stabilizes the negative charge as it is more electronegative than the Nitrogen atom?
An oxygen atom has smaller radius than nitrogen, because of bigger nucleus charge, what contracts orbitals.
Oxygen has higher electronegativity than nitrogen, as p electrons have mutually low nucleus/kernel screening coefficient, so present or incoming electrons are in oxygen atom attracted by higher effective nucleus charge ( plus force is bigger due smaller radius. )
For more, see Slater's rules
Yeah, you have noticed right thing. Well, remember one thing nitrogen has half filled state so it's more or less inert in comparison with it's neighbours i.e oxygen. So it has no need to take electrons because if it takes it that will break it's maximum spin multiplicity which will result a little distabilised state . Where as oxygen is more electronegative has affinity towards electron so it going to take an election but remember the specific example you asked has a annoying point that is (2- ) charge or capturing 2 electrons will create a massive distabilised state because of its small size and inter-elctonic repulsion.