VSEPR Theory

$XeF_2$ has 2 bond pairs and 3 lone pairs on Xe ($sp^3d$, linear); the three lone pairs are in equatorial positions.
So $XeF_2$ has the maximum (3) lone pair–lone pair repulsions among the given molecules.

$NO_2^+$ is linear ($sp$), bond angle $180^\circ$.
Order: $NO_2^+$ ($180^\circ$) > $NO_2$ ($\approx 132^\circ$) > $NO_3^-$ ($120^\circ$) > $NO_2^-$ ($\approx 115^\circ$)

$NH_3$ and $PH_3$: $sp^3$ with one lone pair, both pyramidal.
$XeF_4$: $sp^3d^2$, square planar; $XeO_4$: $sp^3$, tetrahedral — not isostructural.
$SiCl_4$ and $PCl_4^+$: both $sp^3$, tetrahedral.
Diamond and SiC: both have tetrahedral arrangement with $sp^3$ central atoms.

$NO_3^-$ has $sp^2$ hybridisation with no lone pair on N, so it is plane triangular.

$XeF_2$ is $sp^3d$ hybridised with three lone pairs and is linear.
$ICl_2^-$ also has $sp^3d$ hybridisation with three lone pairs on iodine and is linear, so the two are isostructural.

$BF_4^-$: 4 bond pairs, no lone pair, $sp^3$, tetrahedral. $NH_4^+$: 4 bond pairs, no lone pair, $sp^3$, tetrahedral. Same shape and hybridisation.
$BCl_3$ is $sp^2$ trigonal planar, but $BrCl_3$ is $sp^3d$ T-shaped.
$NH_3$ is $sp^3$ pyramidal, but $NO_3^-$ is $sp^2$ trigonal planar.
$NF_3$ is $sp^3$ pyramidal, but $BF_3$ is $sp^2$ trigonal planar.
So $BF_4^-$ and $NH_4^+$ are isostructural.

$PCl_3$: P has 5 valence electrons; 3 form bonds with Cl and 2 remain as one lone pair → 3 bond pairs + 1 lone pair.
$H_2O$: 2 bond pairs + 2 lone pairs.
$BF_3$: 3 bond pairs + 0 lone pairs.
$NH_2^-$: 2 bond pairs + 2 lone pairs.
So the species is $PCl_3$.

In $BrF_3$, Br has 7 valence electrons; 3 form bonds with F and 4 remain as 2 lone pairs. So there are 5 electron pairs (3 bp + 2 lp), and the electron geometry is trigonal bipyramidal ($sp^3d$).
Repulsions decrease in the order: lp–lp $\gt$ lp–bp $\gt$ bp–bp.
In a trigonal bipyramid, an equatorial position has only two neighbours at $90^\circ$, whereas an axial position has three neighbours at $90^\circ$. So lone pairs placed in equatorial positions have fewer strong $90^\circ$ interactions.
With both lone pairs equatorial, they are $120^\circ$ apart (reducing lp–lp repulsion), and each has fewer $90^\circ$ contacts with bond pairs (reducing lp–bp repulsion).
This gives $BrF_3$ its T-shaped (slightly bent T) molecular shape.
So the lone pairs occupy equatorial positions to minimise both lone pair–lone pair and lone pair–bond pair repulsions.

$XeF_2$: Xe has 2 bond pairs and 3 lone pairs ($sp^3d$), so it is linear. $IF_2^-$: I has 2 bond pairs and 3 lone pairs, also linear. They are isostructural.
$NH_3$ is pyramidal (3 bp + 1 lp), while $BF_3$ is trigonal planar (3 bp, no lp).
$CO_3^{2-}$ is trigonal planar, while $SO_3^{2-}$ is pyramidal (S has a lone pair).
$PCl_5$ is trigonal bipyramidal, while $ICl_5$ is square pyramidal (5 bp + 1 lp).
So $XeF_2$ and $IF_2^-$ are isostructural.