Well, the link I provided you was to a software engineer who created a monkey-simulator and produced Shakespeare's work, so unless he somehow faked it, I must trust a real experiment producing the result over the words by any physicist or otherwise.
Years ago, a woman mathematician claimed something about statistics that got all mathematicians in the world upset. They accused her of being wrong, but after some time, they realized she was right. The moral is, only trust a statistician to 99%.
True, but you have to understand the spirit of the analogy of the monkeys rather than fixated on the physical monkeys and actual old typewriters. It was used as an metaphor for true randomness. Instead of saying "if we had a computer that randomly...", the story of the monkeys and the typewriters were just more fun to think about. Most monkey won't actually type randomly on the typewriter but more likely type certain series or favor certain keys. But that wasn't what the analogy was to be meant.
If you have a consistent pattern that repeats, then you don't have true randomness. In an infinite series of random numbers, theoretically, every conceivable number should come up some time, otherwise the series is loaded (like loaded dice) and is not truly random.
Let's say you have the number 0 to 9. You are going to create an infinite series of random picks of them. If only the number 0 to 8 shows up, ever, infinitely, then 9 isn't part of the set of numbers to be picked, so it's not truly random between 0 to 9, but random between 0 to 8. Also, we're supposed to expect an even distribution of the numbers, so after 10 million numbers, we should see each and every number occurring about the same number of times. If not, then it's also loaded. True randomness should be able to produce all number and somewhat evenly distributed. At least that's how I understand randomness.
I'll get to the rest of your post later.
--edit
Before I go on though, my understanding is that Schroeder is a young Earth creationist who's trying to change science to fit 6-day creation in Genesis. And his CV doesn't really impress any more than any other scientist you can read.