The problem of Spain is intense anthropogenic land (mis-)use including deforestation and mining, which led to the desertification especially in the south in the first place; this has been happening since pre-historic times. So, no, there shouldn't be deserts in Spain right now, and they are for the most part not natural. Read further here: https://www.sciencedirect.com/science/article/abs/pii/S00346... (free version available via google search on ResearchGate)
Exactly. Los Monegros was until 1766 Europes largest oak (encina) forest. It had been a communal forest. The forest was privatized and cut to cultivate cereal.
Deforestation is the only major relevant factor that led to the formation of any desert in Spain, and this started a "long" time ago, and is purely and simply controlled by humans. For example, the famous Tabernas desert is the result of deforestation, followed by intensive erosion.