The mulch washes down the sides of the mounds and it's a mess to mow near them. I thought about removing the biggest mound then I thought about building a retaining wall around it.

Not sure what these mounds are for. There are diffent ways you could go, plant them maybe. Do you like them? The simplest thing would be to dump more mulch on top and aim the mower to send flying chips back towards the heap when you mow. If you add more periodically it should stay covered. The current layer is too thin or has started to age out.

My issue is a downspout. I don't think all the water from the downspout would be good for even a short retaining wall. Would the drainage pipe that I've seen in retaining wall how to videos be enough to handle the flow from a good hard rain?

There's a lot of "it depends" in the answer.

Do I make an exit through the wall for the pipe?

If the work involved is reasonable I would consider this a superior solution. If you want to get fancy you could have the water spash into a bowl or something.