Can I use flour to make a carrot soup more thick?
How much flavor the flour adds will depend largely on how long it fries with the veggies.

The longer it fries the nuttier it will taste- until it starts to burn of course. If it doesn't cook for long enough (it doesn't take long) it will taste raw which is not pleasant.

You should cook your veggies until they are done and then add your flour and cook just until it starts to smell nutty. I assume this is similar to what you do with your goulash.

Overall- I think that the flavor of even a fairly dark roux would work well with an onion and carrot soup. You could even use this to salvage your already watery soup by making a roux separately and wisking it in.