Translation:I am taking the dog.
Would I am walking the dog be translated as "Ik loop de hond?"
Almost, that would be Ik loop met de hond or more commonly (and more specifically linked to the task of walking the dog over just walking together with the dog): Ik laat de hond uit.
That sounds like "I let the dog out.", which in English suggests that I'm staying inside.
If you want to say "I let the dog out" (and I'm staying inside). I would say "Ik laat de hond naar buiten", whereas "Ik laat de hond uit" is generally used to say that you are going to walk the dog so it can do its business.
Who let the dogs out?
If you say meenemen it's bringing. For example, I will bring with me the dog, ik zal de hond meenemen?
"I will bring the dog with me" = "Ik zal de hond (met me) meenemen"
However, "I will take the dog with me" is also "Ik zal de hond met me meenemen".
If I were to write this as "ik ben de hond aan het nemen" would that be correct? I don't remember how to add objects in the continuous.
Grammatically speaking you are correct. Just in this context the "continuous" is not used in Dutch.
As El2thek has already sead its grammatically correct but it doesnt sound good
well literly that would be saying: I am taking the dog, people might misunderstand what you want to do to the poor thing...