Learn French: Il est plutôt content.

Not in this exact sentence. Here, plutôt simply modifies content to mean 'rather' or 'fairly'. To express an alternative meaning 'rather than' (which is very similar to au lieu de), you need the phrase plutôt que.

What is the difference between plutôt content and assez content?

They are very close in meaning. Assez means 'quite' or 'fairly'. Plutôt means 'rather', and often carries a slight nuance of contrast to expectations. Saying Il est plutôt content implies he is surprisingly happy about the situation, instead of upset.

Where exactly does plutôt have to go in the sentence?

Because plutôt is an adverb modifying the adjective content, it must go immediately before that adjective. In French, you generally cannot put this kind of modifying adverb at the very end of the sentence.

How would this sentence change if we were talking about a woman?

The adjective content must agree with the subject. You would add an e to make it Elle est plutôt contente. This also changes the pronunciation, because the final 't' sound is now pronounced.