Learn Finnish: Onneksi meillä on sentään ruokaa. - Luckily we at any rate have food.

While kuitenkin neutrally means "however" or "anyway", sentään carries a specific tone of relief or concession, meaning "at least". It highlights a silver lining: even if everything else went wrong, at least we have food. You could grammatically use kuitenkin here, but it would lose that comforting nuance.

Why is sentään placed right in the middle of the sentence instead of at the end?

Attitude and contrast adverbs like sentään, kuitenkin, and siis are typically placed immediately after the finite verb (here, on). Putting sentään at the very end of the sentence would sound quite clunky and unnatural to a Finnish speaker.

I recognize onneksi as "luckily", but what case is that word actually in?

It is the translative case (ending in -ksi) of the noun onni (luck, happiness). Finnish frequently uses the translative case to build adverbs that express a speaker's attitude or evaluation of a situation.

As a quick refresher, why do we use the partitive case for ruokaa (food) instead of the nominative ruoka?

In a possessive "someone has something" construction, the thing you possess must be in the partitive case if it is an uncountable mass noun. This indicates that you have an unspecified quantity or "some" food.