No, monsters do not run out of mana.
Think of monster spells as a probability. Each turn the monster has a probability of casting each spell. That probability will define the frequency of a given spell.
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For an example, a spell with 100% probability will be cast every turn. But a spell with 10% probability will be cast, IN AVERAGE, once every 10 turns.
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Fun fact: although creatures do not have mana, they used to have SKILLS. So, back in 2004 when people would train for hours with creatures, at the end of the day some anomalies could happen, like a Wolf hitting 30+ xD