Ok, your question confused me with the airport and TC as if it is two things not one.
when I connect the ethernet cable to TC, why doesn't modem/router show up in connect to list? Or if wrong list, where should I look, to select, to get it to "see" what it is connected to by ethernet?
Are you using v6 in Lion utility? I am using SL with 5.5 utility. There is no such place that I can find. (You will do better downloading and using v5.6 for Lion anyway).
On the manual setup.. summary page it will show the IP you get on the WAN port. From default settings this will all work properly.. there is nothing you need to do.
Plug ethernet from modem router to TC wan port. When you do the TC will get an IP.
If you open the internet tab, you can see Connect using.. ethernet. Just leave that. and everything else on auto.
For connection sharing, you still need to share a public IP.. although TC will complain you are not using public but private IP.. there is nothing you can do about that. Just set the TC to ignore the error double NAT.
When I began installation, I plugged everything in, then plugged in power. I opened Airport Utility, and followed on-screen instructions, but when selecting "connect to existing network with ethernet cable" my modem/router doesn't appear in the list.
Do not use connect to existing network.. that will bridge the TC which you don't want.
You have to use the existing network as if it is public network. And stuff behind your TC is private.
If I set the TC to bridge mode, won't it "take over" all router activity, and the first group of computers won't be able to connect directly to motorola? (I don't want this first group of computers to be able to connect to the TC).
Also, I don't want the TC to connect to the motorola wirelessly.)
If you set the TC to bridge it will not route at all... it will become a client of the existing network. And all IP addresses will be taken from the Motorola.. you will have one flat network, which is what usually people want.
If you are still struggling.. do this.
1. Tell me the IP address the existing Motorola is passing to the clients. Your one big issues here is making sure they are in fact on two different subnets.
2. As long as condition of 1 is fulfulled this will work at default setting. Reset the TC by holding down the reset for 10sec.
Plug Lan motorola to WAN TC.
Plug a computer into the TC to do setup.
Set up as if you have public IP on wan. Tell TC to ignore double NAT.
Set wireless and TC names as you want.
Now test internet.. on a client computer on the TC. it should work fine. At least for everything that doesn't need interactive connection.
3. You can improve things by using DMZ in the Motorola to the TC. That alleviates at least some of the issues that arise with double NAT.