"Do you like watching movies?"
As a basic rule が should always preceed 好き. のは acts as connector in a way that everything before のは becomes the main topic of the sentence. In this case the "action of watching movies" become the main topic. So both would be correct.
It should be correct. In fact I'm not even sure if は would be correct here. Seems unnatural, no?
観る is technically the correct kanji for watching something like a movie, but even a lot of people in Japan just use 見る since it's easier
Adding の after a verb turns it into a verbal noun, similar to adding -ing or using the infinitive in English. Another example would be "Singing is fun." -> 「うたうのがたのしい。」
This can also be done by adding こと at the end, in the same way: 「うたうことがたのしい。」
I missed the の, and maybe made some other mistakes, I don't know: "あなたは映画を見るが好きですか？"
The "Correct solution" it gave was "えいがを見るのは好きですか？", as above. But it chose to underline the ”き” instead of the ”の”, which doesn't make any sense. There's nothing to correct with ”き”, and everything to correct with ”の”, so far as my answer was concerned.