Spice definition of resistor/capacitor

Keyword is capacitive reactance. Capacitor is used instead of an actual resistor to avoid heat loss. 1M resistor is only to discharge capacitor when not under power (safety measure).

Your circuit is overly complicated, but in essence to power a led from mains input you need to drop most of the voltage on something that acts like a resistor but does not get hot.

The capacitor is not coupled to the other nodes.

In this case, the capacitor is not affecting your circuit, so you needn't model it. By Kirchoff's current law, your circuit can't deliver any current onto the spherical capacitor without also being connected to whatever the spherical capacitor is coupled to (which is probably mostly earth ground).

If you want your circuit to be able to deliver charge to the spherical capacitor, you must also connect it to earth (or whatever ground-like object is near the capacitor). Then, of course, earth becomes a node in your circuit and it's no longer true that the capacitor is not connected ot other nodes of your circuit.

Responding to some comments:

Coupled how?

I was just using the word you used. I assumed you meant coupled by electric field lines between the sphere and the things it might be coupled to.

Capacitance is a measure of the energy stored in the electric field, but surely I could raise a sphere far enough from earth that the field strength is negligible on the ground and still store charge (and energy) on the sphere.

You'd have to transport the charge to the sphere from somewhere. That would form a circuit including that other place.

If the charge comes from somewhere in your circuit (elevated in space along with the sphere), then you already had a charge imbalance between the sphere+circuit system and the earth (however far away), and you'd have already had an electric field developed. Moving the charge from the other circuit elements to the sphere wouldn't appreciably change the field between the sphere+circuit system and the earth, so it wouldn't store any additional energy.

Is using KCL circular reasoning: all the current entering a junction must leave the junction therefore there must be a junction?

KCL can also be applied to any closed surface. If you define a closed surface surrounding the spherical capacitor and the rest of your circuit, then the net current across that surface must be zero. Obviously since you've imposed this surface between the sphere and ground (or whatever) you have to include displacement current.