Airport seems to hang, but wired traffic still gets through

Question

I have a newish Airport that I bought to replace a Time Capsule that died.  Both devices have had troubled childhoods.
The new device, which we have had for about six months seems to have some kind of electronic ADD.  Wired traffic still works but the Wifi dies and the airport will not talk to the airport utility.  If I ignore the bleating of the other household members, stomping around loudly enquiring "Is the wifi down?", standing at my office door hyperventilating, saying to no-one in particular "Well, I guess I will go read a book then" etc et etc, I can keep working because everything I use is wired.  Until I decide to cook dinner and look up the recipe on my iPad, whereupon I recall the earlier histrionics...
The only fix is to restart the Airport Extreme. 
Restarting it presumably clears its DHCP table so it gives out dupllicate addresses when something appears.  And that then causes all kinds of problems.
The network has a sixteen port router, two 8 port routers and an Airport Express that extends the network in our long thin house.  There are about 20 devices that use the network including desktop computers, laptops, iPads, iPhones, file servers, media players, Vmware and Parallels virtual machines and other stuff.  This disrupts the entire household including my work-from-home arrangements and it takes about half an hour to go around the house tracking down all the devices and switching everything off, rebooting the Airport Extreme, then going around the house and restarting it all again. 
So, my questions are:
* Do you have an Airport in this kind of complex environment that works successfully?
* Do you have an Airport that seems to crash in this way (wired connections work, WiFi and Airport Utility response not so much)?
* Do you think that the Airport is up to the job?  Am I being unreasonable in expecting it to work in this environment?
* Is there a history of extended wifi networks causing airport crashes - i.e. is it likely that using the airport express to extend the network is causing the problem?
* Do you have a theory about what might be causing the problem?
It might be the hardware, but the Time Capsule was doing the same thing before it died...  I think it is probably a software problem.  Taking it to the Apple Store is not a preferred option because (a) I have to get an appointment and last time I tried that (my main work laptop had died) I was offered an appointment 4 days in the future and (b) it will probably work just fine (because it always does after I restart it.
So, Community - do what you do best! Opiine, theorize, rant and excoriate.  Lemme hear it.
R

Answer 1

The network has a sixteen port router, two 8 port routers and an Airport Express that extends the network in our long thin house.  There are about 20 devices that use the network including desktop computers, laptops, iPads, iPhones, file servers, media players, Vmware and Parallels virtual machines and other stuff.  This disrupts the entire household including my work-from-home arrangements and it takes about half an hour to go around the house tracking down all the devices and switching everything off, rebooting the Airport Extreme, then going around the house and restarting it all again. 
This puts your question outside the reach of most..
And I think your naming is mixing yourself and others up as well.. I am sure the 16 port router and the 2 x 8 port routers are dumb switches.. least ways I hope so.
The shear complexity of the network is going to make troubleshooting that much harder.
* Do you have an Airport in this kind of complex environment that works successfully?
Yes.. I have a 16 managed switch plus a number of other devices.. like WAPs bridges with a modem router.. plus ethernet to everywhere plus other switches. And it all works fine. I do not have the main Time Capsule in my case as router.. it is bridged. I do a lot of repairs and testing so I keep it bridged for flexibility.
* Do you have an Airport that seems to crash in this way (wired connections work, WiFi and Airport Utility response not so much)?
I seldom have issues because I use Mountain Lion as little as possible and control everything from Snow Leopard.. the last OS Apple made with really stable networking IMHO. I also am selective of firmware on the TC.. when a later one acts up .. which they do, I roll back to the last stable version. 
ML always gives trouble but I force it to connect to something else.. problem solved.
* Do you think that the Airport is up to the job?  Am I being unreasonable in expecting it to work in this environment?
I do think that a single airport might not be ideal.. but it depends on the load.. house layout and bandwidth usage.. and loads of other considerations. Overall the airport I would place somewhere at the top of home / SOHO type gear but below pro equipment.
About where Apple would like to be. But the latest AC model has some serious issues.. and Apple is tardy getting down to writing a reasonable firmware update.. still your story with the same issue with older TC presumably may point the finger elsewhere. 
* Is there a history of extended wifi networks causing airport crashes - i.e. is it likely that using the airport express to extend the network is causing the problem?
You actually haven't given a real picture of the network.
Is the extreme the main router? What kind of modem and broadband do you have?
Exactly how is the express setup.. as that kind of crept in without warning.. we were talking about TC and Extreme. Is it wireless extend or roaming setup?
It should be roaming with all the ethernet you must have.
The latest firmware is a dog.. You haven't given model numbers either so I have no idea how old the gear is.. but I am assuming new.. so this is AC Extreme and dual band Express??
There is nothing you can do for the AC model.. Apple have some issues but are very slow to get a firmware out to fix them. The Express can go back to 7.6.1.. but that is hardly relevant if you have roaming setup..
Please give exact models.. A1xxx from the labels on the routers.
Please give their exact setup.. even a few screen shots.
* Do you have a theory about what might be causing the problem?
Lots but info helps no end.. from your end give the above and we can then suggest alternatives.
Actual screenshots of the setup of the Extreme and Express gives us hugely better picture of where things can go wrong.
BTW your post should have been picked up automatically after 24hours by an Apple rep who simply quotes all the documents you should read but at least gives an answer.. there is nothing wrong with giving a bump after 1day and after 2days as this discussion area moves so fast it might go over the page before one of the usual suspects has a chance to see it.