"My older brother is good at baseball."
I would argue that ending this sentence with "da" rather than "desu" is not incorrect
Why do we use ga after yakyuu here? Shouldn't it be "yakyuu de jouzu" or "yakyuu wa jouzu"?
が is used to express a strength or a weakness. It can't really be explained, as you can't explain why it must be good at and bad at.
It can be explained, and you nearly did. With 'baseball' being a strength (noun!), baseball becomes the grammatical subject of the sentence, rather than the person whose strength it is. Literally this translates to: "As for my brother, baseball is a strength/his forte".
the topic of the sentence is the brother, so it gets the wa, you don't get more than one wa in a sentence. as for ga vs de, it seems like both are ok at least in terms of google search hits.
The brother is indeed the topic - rather than the grammatical subject - of the sentence, but that's unrelated to the number of "は"s in a sentence (which there can be more than one of). Also, the number of Google hits is not an accurate standard for the correctness of a phrase.
As far as I know, the only time you get more than one は is when you're contrasting two different topics. That and starting with a set phrase like 「実は」Are there other patterns I'm forgetting?
I don't disagree with your second point, but number of Google hits is a good heuristic for determining whether or not a particular phrase is in common use, or if one phrase is more commonly used than another as in this case.
I wish all the sound bytes would actually play for the kanji options. Especially for ones like 野球 or 上手 that hadn't been formally taught yet.