Is the dsl modem also a router.. if so is the TC in bridge mode.. you cannot use double NAT and have it reliable.
Bridge one or the other.. if you bridge the modem then use pppoe client in the TC and see how that goes.
To get more accurate assessment of what is going on, you probably should run a cable direct to the modem and get the full line stats which you can post here or on a forum of your ISP.. and also the log of the modem should show if the modem itself is failing.. if it having trouble without anything attached.. which I gather from what you say is happening, then you have a more general issue.
LED light for dsl and internet on the modem can also give some info but a full log is usually better.
The dsl modem is not a router. In fact it is the original dsl modem I got with Earthlink back in 1999. It works great! How do I get "full line stats"? The modem is not failing and as I said above I can do the file transfers if the Mac mini is ethernet attached directly to the modem. The problem only happens when the Time Capsule is connected to the modem and the Macbook air (via wireless to the Time Capsule) or the Mac mini (via ethernet to the Time Capsule) or a Windows laptop (via wireless to the Time Capsule) connect to the internet through the Time Capsule.
Ok,, very unusual now to have pure modems.. they went out.. well even before 1999.. but if you can borrow or legally obtain a modem on trial, I would still have a go.. modems seldom last 10years.. let alone more than that.. and the power supplies certainly don't.
Reset the TC and do the following.
It should of course be in router mode.
Set to pppoe client for connection, unless your ISP uses some other crazy method.. ipoa etc??
Set the TC with full SMB names.. TC name, wireless names, and hard disk name.. all should be short, no spaces, pure alphanumeric.
Lock down the wireless channels and everything else.
See how you go.. if it fails due to the pppoe client.. if the TC is older model.. ie more than 1year since purchase.. go back to 7.5.2 firmware as the latest firmware is full of bugs.