Testing the wiring is an interesting project in and of itself, but there may be a simpler and more effective way to address the overall system, at least before resorting to that: Can you take the DSL modem and a laptop to the service entry point and try it right there, thus eliminating the wiring issue? If it works there, then you work on testing your wiring; if it doesn't the problem would seem to be either with the carrier/drop or with the modem itself.
If you intend to use the mic input mic input to your sound card, it will have a current source or pullup resistor to supply a bias to the microphone to which it was intended.
It might not be good for the speaker to have DC flowing through it.
If your phone is connected to the telco, be very careful. The speaker may run at quite a CM voltage and the DC offset might be -48V (depending which way round it is wired).
Telephones wires are very good at attracting lightning. During a storm high common mode voltages may be on the earpiece.
Stricly speaking you should use an audio transformer to couple the earpiece signal into the sound card, this provides DC isolation and some protection against lightning. Adding a TVS across the output of that transformer would be a good idea too.
The ac signal on the earpiece might be quite small. Only a few hundred mV. For a mic input, this will be fine (and can be potted down at the output of your transformer) but for line levels, which are usually 1Vrms it might be very quiet.
Not all phones work the same way, so measure the voltage first using a scope and judge the divsor as required.
The audio transformer should also be protected against DC usign a DC blocking (aka coupling) capacitor.
This circuit protects against discharge and DC currents, though use with caution as a mistake will cost you a PC!
Best Answer
This question cannot be answered without a lot more information regarding the multi-line phones.
Many later model business telephone lines use different interface protocols between the phones and control unit back in the telephone equipment room than the typical "tip and ring" connections used by a "cheap two line phone".
The business phones may be using digital signalling or a combination of digital and analog signalling. The the "cheap" phone will be using the analog type of signalling.