1) Yes, brains have lots of computational power, but you've already accounted for that when you said "human-level AI" in your claim. A human level AI will, with high probability, run at 2x human speed in 18 months, due to Moore's law, even if we can't find any optimizations. This speedup by itself is probably sufficient to get a (slow-moving) intelligence explosion.
2) It's not read access that makes a major difference, it's write access. Biological humans probably will never have write access to biological brains. Simulated brains or AGIs probably will have or be able to get write access to their own brain. Also, DNA is not the source code to your brain, it's the source code to the robot that builds your brain. It's probably not the best tool for understanding the algorithms that make the brain function.
3) As said elsewhere, the question is whether the speed at which you can pick the low hanging fruit dominates the speed at which increased intelligence makes additional fruit low-hanging. I don't think this has an obviously correct answer either way.
1) I expect to see AI with human-level thought but 100x as slow as you or I first. Moore's law will probably run out sooner than we get AI, and these days Moore's law is giving us more cores, not faster ones.
Claim: The first human-level AIs are not likely to undergo an intelligence explosion.
1) Brains have a ton of computational power: ~86 billion neurons and trillions of connections between them. Unless there's a "shortcut" to intelligence, we won't be able to efficiently simulate a brain for a long time. http://io9.com/this-computer-took-40-minutes-to-simulate-one-second-of-1043288954 describes one of the largest computers in the world simulating 1s of brain activity in 40m (i.e. this "AI" would think 2400 times slower than you or me). The first AIs are not likely to be fast thinkers.
2) Being able to read your own source code does not mean you can self-modify. You know that you're made of DNA. You can even get your own "source code" for a few thousand dollars. No humans have successfully self-modified into an intelligence explosion; the idea seems laughable.
3) Self-improvement is not like compound interest: if an AI comes up with an idea to modify it's source code to make it smarter, that doesn't automatically mean it will have a new idea tomorrow. In fact, as it picks off low-hanging fruit, new ideas will probably be harder and harder to think of. There's no guarantee that "how smart the AI is" will keep up with "how hard it is to think of ways to make the AI smarter"; to me, it seems very unlikely.