Best configuration for Time Capsule & Airport Extreme running one wireless network?

Question

We just moved to a house that does not have Ethernet pre-wired so I've had to re-setup the wireless network in a different configuration and it seems we're experiencing deteriorated wireless signal for devices like laptops.  It will work perfectly fine (and fast! wheee!!!) but then it will slow to a crawl and mostly timeout.  before I reset everything and start from scratch, I'd like some help.  As an FYI, when connected via Ethernet to either of these, Internet is blazingly fast… so my problem here is the Wireless portion. 
As some background, here is how we had the network set up in the previous house which was working great (although it was wired for Ethernet which made things A LOT easier):
Airport Extreme (4th Generation)
Wireless Mode: Create a Wireless Network
Internet Facing (Connected to the Cable Modem)
Location: Upstairs Utility Closet (with Smartbox and cable modem)
Time Capsule (1st Generation)
Wireless Mode: Extend a Wireless Network
Joined to existing network above, but also connected to the Ethernet jack
Location: Upstairs Office
Here is the current set up we have running:
Time Capsule (1st Generation)
Wireless Mode: Create a Wireless Network
Internet Facing (Connected to the Cable Modem)
Radio Mode: 802.11n (b/g compatible)
Location: Upstairs Office
Airport Extreme (4th Generation)
Wireless Mode: Extend a Wireless Network
Joined to existing network above with NO incoming Ethernet connection
Location: Downstairs Family Room
My understanding is that the Airport Extreme should pick up and relay the signal downstairs, as well as provide the router for the devices downstairs that need Ethernet (TiVo, Apple TV, etc).
So, my question is (and thank you in advance for helping me sort this out!), what would be the best configuration to optimize the Wireless performance in the house? 
Also, for whichever device is the one picking up the existing network and providing the router capabilities for the Tivo and such, is it possible for it to pick up but not rebroadcast the signal?  I think in this new house we may have a good enough signal from the office, without the need to rebroadcast it.  My theory is that in this house, they are too close together and interfere with each other and/or devices aren't sure which to choose so they bounce back and forth endlessly.  Again, just a theory.
Many thanks again for your help! If I've left out any information, please ask and I can provide. 

Answer 1

Should they be running on the same channel (as with the "Extend" option previously? Or just both on Automatic channels?
I would recommend the Automatic setting on both devices because this will allow each router to scan and choose an open channel automatically. If  you want to set the channels manually, keep them separated by at least 4-5 channels. You would not want to ever use the same channel for both devices in this type of "roaming" setup.
Is there a way to limit bandwidth per device (or action) so that it doesn't monopolize the network and keep other devices from using it?
Nothing that I am aware of. It sounds like  you have some young users who are likely visiting Bit Torrent sites, which will really gobble up the bandwidth with big, heavy downloads. The Time Machine has to back up the same files again when it makes the next pass, so that clogs things up again.
Backups would go 3-5 times faster, on average, if the computer(s) could connect using Ethernet, but I understand that wires are not cool with the younger crowd.
should BOTH the TC and AE be set up in Bridge mode?
The AirPort Extreme should be set up in Bridge Mode.
The Time Capsule...it depends. If the TC is connected to a simple cable modem...then the correct setting for Connection Sharing would be "Share a public IP address" on the TC. That is because a simple modem has only 1 IP address to deliver. The TC takes that address and "shares" it with other devices by setting up a local network for all your other devices.
If the TC is connected to a gateway....a combination modem/router in the same box that usually has 3-4 Ethernet ports on the device....then that device would be considered the main router on the network and the TC would be  configured in Bridge Mode in that case to function correctly on the network.