Power demand beyond maximum deliverable power

The main things that you need to look at when selecting capacitors for a power supply is the voltage rating of the capacitor, if it needs to be polarized, and the equivalent series resistance.

The datasheet of the IC that you are using will specify what value of capacitor you need and many times will specify the actual type of capacitor required.

Electrolytic Caps are polarized which doesn't work well for AC signal, but is the cheapest way to get a high value capacitance, which is why they are used a lot in power supplies.

So, in short, yes, any cap should work OK as long as you are in the allowable voltage range.

A conventional current transformer will do what you want.
As Jim Dearden says, you probably do not need or want a current step up transformer as voltage will be reduced AND the device will interfere with normal circuit action.

As shown below, if you use a conventional current transformer and feed the main circuit via a single turn "winding" on the core, then you will get an output voltage at lower current. Terminate the winding with a resistor and then use the voltage across the resistor as an energy source. The single turn usually consists of a conductor passed through the core laminations. This circuit produces voltage proportional to load current - so at low currents the voltage may be too low to be useful with a regulator. Replace the regulator with an "energy harvester" if you want to use a very wide range of voltages. If you remove R_CT_LOAD and place a minimum load on the regulator output it will work at low currents but Vin may rise to high values for heavy mains load. A dynamic load scheme could be 'easily enough' devised so that Vout was acceptable under most mains-load conditions.

^{simulate this circuit – Schematic created using CircuitLab}

R_CT_Load OR a consistent minimum output load on the regulator is absolutely essential at all times. The resistor value is based on manufacturer's data for the CT used and is designed to give a target V when target current flows in the input single turn.

Note that with any current transformer, if you run it unloaded Vout can rise to very high values and may destroy the transformer or connected circuitry.