"There is a dog."
が is used when information about a subject is important or situationally new to the listener. は is used once the same subject has been mentioned for the sentences following it. I believe は is also used when the subject word is before the question word, and が is used when the question word is before the subject.
It's been a while since I've done whichever lesson these questions are in so I don't know what the cat sentence said, but for this question, it's just stating the existence of a dog or multiple dogs. "There is a dog", "A dog exists", "There are dogs" should all be accurate translations of this question. There's no counter here to tell you exactly how many dogs there are, so in a conversation you'd just have to determine the meaning via context, since「犬がいます」states an indefinite number of dogs.
"There is a cat" and "There is one cat" are basically identical sentences in English. If the Japanese sentence had a counter for a single cat, both should be correct, but "There are cats" wouldn't be a correct translation since it's stating a definite amount.
I think your sentence is more along the lines of "There's a dog there", specifically near the listener, but I'm pretty sure this question (er, statement?) is just trying to convey that a dog exists, not where it exists.
English is weird though… "There is a dog" CAN also mean "There is a dog there" if you were to emphasize the the first word. For example maybe someone was looking for any dog, and they finally found one, they might say "There's a dog!" and point at the dog, which would imply "There is a dog (over there)!" or perhaps "(Over) There is a dog!". But plainly saying "There is a dog" without any context/emphasis at all means you're saying a dog exists in general.
So to answer your question, I think your Japanese sentence is proper, but contains too much extra context that's not a part of the expected answer.