Learn Russian: Я не давал тебе никакого повода ревновать. - I didn't give you any reason to be jealous.

This is the genitive of negation! When you negate a transitive verb like давал (gave), its direct object often switches from the accusative to the genitive case. This is especially common with abstract nouns like повод (reason) and is practically mandatory when the object is modified by a negative word like никакой.

The English translation says 'didn't give... any', but the Russian uses не давал... никакого. Is this a double negative?

Yes, and Russian loves double negatives. Whenever you use a negative pronoun or adjective starting with ни- (like никакого, which means 'no' or 'not any'), you are required by Russian grammar to also negate the verb with не.

Could I use the perfective verb дал instead of the imperfective давал?

You could, but it changes the nuance. Using the imperfective не давал emphasizes a continuous reality over time: 'I have never given you a reason [throughout our relationship].' The perfective не дал would focus on a single, isolated incident: 'I didn't give you a reason [during that specific argument].'

Why use повод here instead of причина for the word 'reason'?

While both mean 'reason', they are used differently. Причина refers to the deep, factual cause of something. Повод refers to a pretext, an occasion, or 'grounds' that trigger a reaction. When talking about giving someone an excuse to feel an emotion like jealousy, повод is the standard word.