"Do you like sports?"
は indicates the topic, which is sports. I think you've confused the two grammatical terms? In "Do you like sports?" the subject of the sentence is you. Subject/verb agreement checks can help. If you change the subject of a sentence the verb has to 'agree'. If we replace 'sports' with 'cats' in "Do you like sports? " the verb remains the same: do you like cats? However, the verb 'do' changes to 'does' if we switch out 'you' for 'John': does John like sports? Japanese doesn't have such subject/verb agreement which allows A subject can be left out/implied in Japanese, but a sentence is always about something; a topic. Very often the subject of a sentence is also the topic which is why many confuse the two.
Japanese grammar is different from English grammar. No language will have the same grammar or order of placement. When studying another language, after the phrase or sentence has been translated, you need to change the order or add a few words to make it valid in the language it is being translated to.
That's a totally understandable misconception. は and が would both make sense with 好き, but the meaning is different; I can't really explain it well, since I'm an amateur myself, but:
は is the "topic" particle, and が is the "subject" particle. So, my take on that is, if you're discussing sports in general, you'd say スポーツは好き, but if sports come up and you want to state your stance on them, you'd say スポーツが好き.
Another example; if someone asked you if you liked sports, you might reply: いいえ、 スポーツが好きじゃないですでもチェスは好きです。
There is no direct object relationship between "sports (スポーツ)" and the adjective "likeable (好き)." It is grammatically incorrect to use を with an adjective.
Having said that, an increasing number of people use を with 好き, and perhaps in the future it might be treated similarly as the ～たい form or the potential form - using を to replace が when が marks the target of preference or ability.