It's difficult to answer the question about capacitors as it's so difficult to follow the second diagram, but it looks to me as if they are doing a different job. However, I think that they should all have one end to ground. You can probably use the capacitors in both diagrams - in the first, they are power supply decoupling caps which are a good idea, and in the second probably to reduce noise and small random fluctuations on the output which you often get when a variable resistor is involved.

As for power supply - the original diagram shows +7V and -7V. You could replace these with +12V and -12V from an ATX supply. That should work fine and is well within the 741 ratings. You can't use just +12V and ground; the 741 would be happy but the output would not go right down to zero.