What is the concentration of hydronium ions after a 100ml solution of 0.02m magnesium hydroxide and 100ml of 0.05m nitric acid?

That is the balanced chemical equation.

Before reaction,
milli-moles of Mg(OH)2 = 100 x 0.02 = 2
milli-moles of HNO3 = 100 x 0.05 = 5

After reaction:
Mg(OH)2 is limiting reagent. It will react completely leaving some HNO3 behind.
milli-moles of HNO3 react = 2 x coefficient of HNO3 = 2 x 2 = 4

milli-moles of HNO3 left = 5 -4 = 1

Now, total volume = 100+ 100 = 200 mL

It is asked for hydronium ion concentration, which is basically hydrogen ion conc.
 1 HNO3 will dissociate to give 1 H+ ion.

so, milli-moles of H+ = 1 

So, concentration of H+  or hydronium(H3O+) = 1 / 200 = 0.005

Which of the following did you identify correctly