Just to be sure of your setup:
The phone company side of the line should be a single pair of wires.

They took this pair to a filter and split it into 2 lines, one DSL
filtered (for voice) and one not (data). You now have two pairs of
wires

Exactly.

The distribution block in your basment should have some sort of
terminal connections for those 2 pairs.
Most of the feeds for the house will go to the "filtered" side.
The feed for your computer room will go to the "data" side.

Close. The only difference is that the existing distribution block only
deals with lines for voice. The data line goes its own way.

Are the feeds single pair, or double (or more) pair?

Not so sure what you mean by "feeds"?

If it is the incoming phone line from the phone company, it only has
one cable that has 6 pairs inside. Only one pair is being used. The
rest are cut off and rolled up.

If the feeds that you are referring to are the phone lines leading from
the distribution block to phone outlets, they are all either cat-1 or
cat-3 cables that have 2 pairs in each cable (there is one cat-3 with 4
pairs).

The lines that I want to extend are the lines leading to phone outlets.
I need to move the distribution block to a more convenient location;
but the phone lines are too short. I will use the
twist-solder-and-heat-shrink-tubs approach to attach an extension to
each line line.

Or, you can do what we did and go wireless. (modem and access point
sit at a single station and all computers have access.

I think you are referring to wireless computer connection, not cordless
phone connection.

Wireless should be fine for internet connection. But I need the
computer connection to stream video. The current wireless standard is
simply not good enough for video according to people who have tried
this. Even the A+G standard is just the minimum, not for the best
result (according to what I read from Microsoft web site related to
their Media Center Extender such as those from LinkSys and HP).
Therefore, I have chosen the safe-bet, and installed wire through out
most of my house. This whole house wiring project will be done as soon
as I am done with the phone line connection (I am not familiar with
phone wiring). This is the thing that is holding the project back.

The latest pre-N wireless standard may be quite good for streaming
video. But I am not very sure. I guess we will hear more about the
pre-N and the final-N standards in the coming years.
Thanks for sharing your thought.

Jay Chan