1) starting from the "Working configuration", unplug cap and LED, plug in 5v from arduino = unresponsive.
You are saying if you power up the chips they stop working
Is the ground common?
Are you, perhaps, drawing too much power from the Arduino board?
So, find a separate power supply (eg. a 2 amp wall-wart) and use that to supply +5V to the Vcc of the shift registers.
This is called paricitic powering and is a good way to destroy chips. Unless you connect all the chips to the 5V line you don't stand a hope in hell of getting anything working. Chips are made to take power from their 5V input and ground.As you try and scale up you get two problems, first you get noise on the supply lines that distributing decoupling capacitors can help with. Second the signals that go to all the chips, like the clock signal, are not strong enough to power all the chips.