1) I'm a US native speaker and this is the first time I've ever seen the word "ironize." And at first glance I didn't understand it, and thought it was a typographical error for "ionize." I see that it's in the American Heritage Dictionary online, but not my decade-old print copy. It must be a fairly new word. Yes, I think it's best to avoid it. 2) My suggestion is: "Moreover, he also makes use of sarcasm in headlines such as 'Reyes Magos y Reinas Magas' ('The Three Wise Men and Women'), an ironic reference to the absence of women in the cavalcade of the Three Wise Men." 3) The distinctions between sarcasm and irony are subtle (and often debated). To me, irony is a kind of sarcasm. Sarcasm means language meant to hurt someone's feelings. Irony means being sarcastic by saying the opposite of what is meant. It might not be necessary to include both words in the same sentence. One might say: "Moreover, he uses ironic headlines, such as 'Reyes Magos y Reinas Magas' ('The Three Wise Men and Women'), which points out the absence of women in the cavalcade of the Three Wise Men." That actually is three words shorter than the original.

Personally I think it's quite unusual to use the verb 'to ironize' and that people normally say something like "which presents/shows/portrays the role of women in an ironic way" ... But I just googled 'ironize' and I see that Google says the use of this word is growing - and I note the Americanised spelling - so perhaps it's more common in American English?

L.K, This is the first time I have heard this verb as well. The much more common verb of that type that could be used there is “satirizes”. Another way to phrase it might be “which critiques the role of women”.