Time Capsule as a wireless repeater?

I would be placing the new Airport device in the adjacent room where my existing ADSL modem is located, can I connect these two devices wirelessly to extend my Wi-Fi range in the house?

Sorry, but no. There are an extremely few non-AirPort routers that are WDS-compatible with the AirPorts. Most likely, your Linksys is not one of them.

In addition to Tesserax's comments, you should be able to configure the Time Capsule to "join" your existing wireless network and backups would occur over wireless. The Time Capsule does not "extend" or "boost" the wireless network in this type of configuration and the ethernet ports will not be active.

It would be better and backups would occur much more quickly if you could connect the Linksys and Time Capsule together using an ethernet cable.

Is that a possiblility?

1) Why will its ethernet ports be not active?

Basically, because when configured to "join" a network, the TC will be performing as a wireless client ... and not a router. Therefore, its Ethernet ports will not be active.

2) Isn't it extending my Wi-Fi range when used it like this?

Again, no, not if it's joining it.

2) So if TC is joining my existing network, my wireless device sitting next to TC will not connect to TC but the linksys, right?

That is correct, your wireless devices, including the TC, will be connecting to the Linksys.

As a wireless client, in addition to backups, you could use the TC for: 1) Use its internal hard drive as a NAS, 2) Attach an external USB printer and use it as a print server, and/or 3) Attach an external hard drive and use it as a file server.