Can Time Capsule act as WI-FI access point connected to router over ethernet?

Network Mode: changed to extend a network

This is wrong.. you must use create a wireless network.

Using Extend wireless only works with another Airport router and you MUST NOT connect by ethernet at the same time.

Wireless Network Name: Matching

Wireless Password: Matching

This is correct.

You can easily tell which client is connected.. open airport utility and check the wireless client list. Click the icon of the airport.. and in the summary roll your mouse point over the wireless client and a bunch of new info will pop up.

In the client you can also check which AP is being used.. hold down the option key on a Mac and click the wifi fan in the top menu.

This will give you the wireless stats from the computer end but also show BSSID which is the actual MAC address of the wireless in the AP.

So you can see good agreement between the Time Capsule which is working as AP and the Computer.

You can confirm BSSID or MAC address by holding down the option key and double clicking the Airport in the Airport Utility.

For your main router you will need to dig into the interface. Usually details are available on most routers under advanced wireless or network or LAN.

Yes.. easy.

Start with a factory reset is easiest..

Use the Airport Utility in your Mac or iOS device.. I recommend using a simple name for the TC, no spaces and pure alphanumerics.

Now you can set the name of the wireless network to match the ISP router.

You need to do this manually.. if the main router was another Airport it would be automatic.

There are several issues.. particularly roaming does not work particularly well between routers from different manufacturers on some devices.. if you have problems you will need to put your phone etc into airplane mode and then out again so it picks up the nearby AP.

In addition to setting up the Time Capsule to use the exact same wireless network name (SSID) that the ISP modem/router is using........the Time Capsule will also need to be set up to use the same wireless network password that the ISP modem/router is using.

So, both the ISP modem/router and the Time Capsule will broadcast the same wireless network name using the same password.

I don't suppose there is anything I can do on the router / access point to improve this roaming issue or does it all come down to the device itself?

Nothing that you can do on any of the access points. Try temporarily turning off the WiFi on the PC when you move it from an area near one access point to another area near another access point. Then, turn the WiFi back on at the PC and it will usually then pick up the signal from the nearby access point.

Same deal for iPhones and iPads, although some of the new iPhones and iPads seem to do a better job of automatically switching to pick up the signal from the closest access point.