Here's another way to answer the question of where the information in DNA came from: the environment. Most people would agree that the Grand Canyon wasn't created directly by an intelligence, even if they believe in God; rather they would agree that it was carved out by water over a long period of time.
The information in DNA is a set of blueprints which were refined in a sequence of organisms living over billions of years. As those organisms reproduced, generational mutations in their genetic code were shaped by the environment in the direction of what helped those organisms survive. The ones whose mutations were not beneficial died out.
So rather than it being an amazing thing that all this "carefully designed" data exists in the form of DNA, it's actually a natural result of the way our universe is built, every bit as natural as the way the Grand Canyon formed. The information, the complexity of that data, is just the result of long years of interaction with the outside world, much like the steady accumulation of wrinkles on our face as we age.
That's not to say that it isn't amazing that the universe accumulates order and assembles information like it does. I do think that's pretty amazing. But if it didn't, we wouldn't be here to discuss it. So if there have been a billion universes before this one, or if they exist alongside us, but they do not tend towards order like ours does, then they're just a whole bunch of non-living noise.
There's no one alive in those chaotic universes to propose that they must have been created because "Look at all this complex stuff, it must have been designed". Our universe has just enough order to allow us to exist and contemplate ourselves, but just enough disorder to show that this is not a carefully-designed universe made by a loving, attentive God.