I live in an apartment building. Around the 4th floor. My shack is in
one room and the nearest Cold Water Pipe is in the bathroom.

It would take about 20 feet of wire to go around the wall to reach the
little Cold Water pipe in the bathroom under the sink.

I am not sure which gauge of wire to buy, or must it be 100% copper?

As for clamps: What would be a good type to use? The largest thing to
clamp would be a PL259 coax connector.

What about soldering each clamps to the wire?

I have a few radios & things and maybe even an antenna that I want to
bring to a common ground.

Is their some kind of copper pipe that I could have in a horizontal
position behind the radios on my desk area, everything then clamp down
to the horizontal copper pipe, then to the 20 feet wire, then to the
Cold Water Pipe?

Is their a meter that I could also have in this line, to watch any
grounding fluctuation? I think that would be fun to watch!

73, SR!