Learn German: Ich missverstehe diese Geste oft. - I often misunderstand this gesture.

No, the prefix miss- is inseparable. Even though it might remind you of separable prefixes, it stays firmly attached to the verb in the present tense. That is why we say Ich missverstehe instead of putting miss at the very end of the sentence.

How would this sentence look in the spoken past (Perfekt)? Do I add a ge-?

It would be: Ich habe diese Geste oft missverstanden. Because miss- is an inseparable prefix (and so is the ver- part), you do not add the typical ge- to form the past participle.

Why is it diese Geste and not diesen Geste?

Geste (gesture) is a feminine noun, and it acts as the direct object of the sentence, meaning it takes the accusative case. For feminine nouns in the accusative, the ending for demonstrative words is -e, giving us diese Geste.

Why does the adverb oft come at the very end of the sentence?

In German, adverbs of frequency like oft usually come after a definite direct object (an object introduced by words like der, mein, or diese). That is why oft follows diese Geste rather than coming before it.