Acidity order of organic compouds
Okay, so if we talk about the acidic strength of a compound, consider adding a base to a solution of all the above. Practically, a reaction is bound to take place-if possible-due to thermodynamic reasons. Hence, reaction in IV will take place since the base will try to find the most acidic Hydrogen, which is why one should consider the removal of H atom from which resonance can take place. 

Now for the order of acidic strength, I, II and III are individually more acidic than IV, due to extensive conjugation in all three but the last. This consequently makes the conjugate bases stable and acids stronger. Now for the order among these, notice that Cross-conjugation takes place in III, as we have COO- group. Hence, III will consequently be less stable and a weaker acid.

The order and reason stated by you for II and I are correct, and hence the order for acidic strength is:

II>I>III>IV

i.e. option A.