"There is a dog."
You use います because a dog is animate. あります is used to say that there is an inanimate object. So for example:
人がいます: There is a person
猫がいます: There is a cat
木があります: There is a tree
コンビニがあります: There is a convenience store
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.
I entered 犬がとります but duo did not accept it. Is it correct to say that provided we were talking about a small dog? Say, a Yorkshire Terrier or something along those lines...
とります is a completely different verb meaning "to take." You would use います for the "to exist" verb for any breed of dog, or any animate being or animal. 犬 can also refer to any dog, regardless of size or breed.
I think I saw a different section talk about how there's different COUNTING words for animals based on size category, but that wouldn't apply to imasu. That's probably where you were getting confused.