It is possible to charge a capacitor, then detach it from the circuit, and carry it around. It holds the charge (if it's healthy), very much like a battery. In that sense it need not be in a loop.
Later you can connect the capacitor back into the circuit (or to some other circuit), and it discharges through neighboring components.
What function does the circuit in post #3 carry out? What is on the other side of out1, out2 and out3?
It looks like it could be a triple ramp or sawtooth (or similar) generator if V1, V2 and V3 are all squarewaves.
If you could post links to, or images of, just a couple more or a few more of the capacitor loop schematics you saw online, this thread will probably move forward a lot quicker. Thank you.
A forum is not meant to help only one person, but it is meant to help many people with similar problems.
Thus you should say what helped you to solve the problem.