If you said 돈이 있어요 then yes. But since it is 돈을 you have to say 가지고 있어요. 가지고 있다 Also means to have/be holding something.
"저는 한국 돈이 많이 있어요" is also perfectly correct but none of all Korean people say" 저는 한국 돈을 많이 있어요."
Why is this in present progressive? To me this translates to "I am carrying a lot of money."
That's technically what it is saying but a more natural way to translate into English is I have a lot of Korean money
Just as word meanings and word orders don't always map 1:1 between two languages, neither do verb forms always map 1:1.
In what circumstances would you use 가지고 있어요 rather than just 있어요? Is 가지고 있어 used more when talking about money?
가지고 있어 is emphasizing that the thing (here in this case, the money) is on your body, you're really bringing it with you. While 있어 just means you have it, and does not really tell us the fact that you bring it with you now or not. Hope this helps ;)