Why are so many different organophosphates able to deactivate acetylcholinesterase?
Acetylcholine's normal function is to cleave acetylcholine's ester bond to convert it into acetic acid and choline, but certain compounds can interfere. The short answer is that these nerve agents all share similar structural components allowing them to be bound by acetylcholinesterase, where they then either react to alter the enzyme's binding site, preventing activity, or are just very slow to hydrolyze, temporarily inhibiting the enzyme until it can remove the compound. 

Not all of these related acetylcholinesterase inhibitors work in exactly the same way, but if you examine their structures and compare them with acetylcholine, the normal substrate of the enzyme, you'll notice some similarities:

Acetylcholine: 

Sarin: 

Tricresyl phosphate: 

You can see that they all have this central double-bonded oxygen (a few have sulfur instead, but are converted to oxygen metabolically), which is important for fitting into the enzyme. The ester bond holding the choline to the carbonyl in acetylcholine is relatively easily hydrolized, but phosphoesters are much slower to hydrolyze. From what I can tell, this seems to be one mechanism behind trecresyl phosphate's toxicity and probably why it's not as acutely toxic as something like sarin or VX. Sarin, on the other hand, has a pretty good fluoride leaving group. This lets it form a strong covalent bond with a serine's sidechain in ACE's binding pocket. This reaction is effectively irreversible, making sarin very dangerous as it totally disables any ACE it reacts with, meaning it takes very little to cause a significant effect.