Time Capsule - Can it run as DHCP router, create a wireless network and plug into switch for wired network all at the same time?

Question

I have a closet with Wired connections from all over the house terminating in it. I also have the internet modem and a switch in this closet. My old setup was a netgear in the closet creating a wireless network and also feeding the Ethernet ports all over the house. In my room, I had an Airport express (a month old) bridging the wifi to make the wifi network larger. The Express was hard wired from the wall and my mac mini was using it's wifi (printer was plugged into other ethernet port). Everything was working great.
Fast forward to last week. My Netgear died. So, I bought a time capsule. My plan was to move the Express to the closet and have it create the wifi network and also then run cat 5 cord from express to the switch to feed all the wired wall plugs. Reset it and set it up to create the wifi network as well as DCHP NAT. Wireless was running great on it. Then I plugged in the Cat 5 in the second Ethernet Port to the switch. Wifi stopped working and none of the computers would connect using wired connections either. They would see a 'network' but nothing would work. I would unplug the Cat 5 feeding the swtich and the wireless would start working again. Plug the Cat 5 back in and wireless would stop.
So I thought I would try the Time Capsule in the closet and see if I could get it to work. Pretty much the same thing, except at this point I figured out if I went in and told it NOT to create a wifi network the wired portion worked fine and all the wired computers in the house worked fine. Of course, then I didn't have Wifi.
I'm not clueless when it comes to networking, but I can NOT figure this out! Can either the Time Capsule or Express Create the wifi network AND also feed the data to the switch for all the wired computers?
To simplify this is what I want my end result to be. I prefer the Time Capsule to be the wireless connection extender, but if I have to have the Express do it, I'm fine with that.
1. Express in the closet creating the wifi network as well as feeding the data to the wired connections via my switch.
2. have the Time capsule in my room at the opposite end of the house plugged into the hard wired connection to extend the wrieless network as well as have wired connections such as my printer and my Mac Mini.
If I can get #1 to work, I don't think I'll have any problems with #2. Help!

Answer 1

I have an equally complex setup with a managed switch actually but I have never seen the TC have this kind of issue.
What I do notice is you have changed the TC from defaults.. you are using a non-default IP range.
I have run across issues like this where people move things from default.
I would like you to start over.. factory reset the TC. That will set it back to default router mode.
Do the min setup on the TC..
1. Change all names from what you were using. That includes the TC name and wireless name/s
Make them short, no spaces and pure alphanumeric because that is the correct way to network. Apple default names with spaces and apostrophes are fundamentally bad.
2. With just the TC plugged into the switch.. and a single computer connected by ethernet. Power cycle the whole network.. this allows the switch to clear all the old MAC address.
3. Test on the computer just ethernet. Turn the wireless off.
The computer must use dhcp and must get an ip from the TC.
It has to get the correct Router.. ie Gateway.. I wish apple could stick with proper network terminology.
And it must get same address or valid DNS server address/es.
If it fails.. please plug ethernet directly to the TC.. bypassing the switch.
You should get the standard IP addressing and internet connection.
If not please post the screenshot from the Mac of the network preferences showing the ethernet setup.
I need to see what address it does get. I need you to make sure the ethernet is the TOP of the list. So rearrange the order of network connection. With the airport off it should just go to the top of the list.
If it appears to be working .. but a browser cannot connect, open a terminal and ping an internet address.
Ping the actual gateway address the ISP gives you and ping the ISP dns address.