I'd never heard of this Douglas Axe, so I went and looked. He apparently works for something called the Biologic Institute, funded by the Discovery Institute. Interestingly enough, he has a PhD in chemical engineering. So you will have to excuse me if I don't accept him as an authority on either evolutionary biology or probability theory.
It would be one thing if he were actually an expert on evolutionary biology or probability theory (preferably both), but using a paper written by a chemical engineer to argue that macro-evolution is too 'improbable'? I can only conclude that this is an argumetium ad verecundiam (argument from authority) logical fallacy.
Furthermore, his actual paper attempts to use mathematical equations to 'prove' that complex adaptations would have been limited to between two and six base changes in the lifetime of the Earth. All this proves is that the equations he selected have those limitations; it does not prove that evolutionary adaptation is so limited. The most he could have shown is that the question is still not settled; he certainly did not show that adaptations requiring more than six base changes would have been impossible.
That completely leaves aside the fact that organizations like the Discovery Institute are focused on trying to disprove evolutionary theory to replace it with an evolved version of an ancient creation story. Yet even if they somehow managed to actually disprove evolutionary theory (something they have not even come close to doing, despite all the dust and smoke they kick up), it would not establish that their chosen alternative was true. Not at all. That means they're practicing a bait-and-switch tactic, rather than practicing actual science with the intent of discovering anything.
If they ever managed to disprove evolution, I would wager real money - all of my income for the rest of my life - that they would not spend any real time or effort demonstrating that their alternative had any scientific value. Instead, it would be presented as a fait accompli, something that was true because evolution was supposedly false, even though this is a false equivocation. Even if BibleStudent were right that evolutionary biologists were practicing "blatant lies" and "outright deceit", it would in no way justify the Discovery Institute and other such organizations doing the same thing to 'prove' their own beliefs.
Also note that his claims aboug evolutionary biologists being deceitful liars has in no way been proven and is thus an argumentum ad hominem (argument at the person) logical fallacy. The only thing he has succeeded in demonstrating is that he his blatantly prejudiced against evolutionary biology to the point of considering its proponents to be deceitful liars. This accomplishes nothing.
What matters is whether creationist organizations such as the Discovery Institute can present evidence that conclusively shows that their alternative, creationism, is more valid than evolutionary biology. This does not mean presenting bad arguments (such as the ones that Axe expounds on in the paper that BibleStudent linked) to try to weaken evolutionary biology to the point where they can supplant it with creationism. The reason this does not work is that even if evolutionary biology were in fact weakened to that point, it would not mean that scientists would automatically flock to creationist ideas. Instead, they would examine the actual evidence and come up with an alternative theory that worked better.
Realistically, what is far more likely is that someone will eventually come up with an explanation that explains the things that evolutionary biology does not yet explain well, as Peter Higgs did with the Higgs field, and thus the new explanation would incorporate evolutionary biology into itself. Comparatively, the idea of creationist "intelligent design" cannot even come close to providing the explanatory power for how life on Earth came to be so diverse. It can only propose that organisms were made to order by some higher entity, which explains practically nothing of value.