To trip when current is in excess of 15 amps and protect the circuit components.
Replace the 30 Amp Breaker with a 15 Amp breaker.
The maximum current rating for a 15 amp circuit breaker is 15 amps.
Not unless you change the wiring for that circuit. The breaker protects the wiring and if you install a 40 amp breaker on a 15 amp wire circuit you will have a fire in your home.
A 15 amp dedicated circuit breaker should be used for the dishwasher. The disposal can be wired to the general kitchen 15 amp circuit.
No tandem breakers are of the same value.
The pool light is usually on a 15 amp circuit. The breaker feeding this circuit must have a GFCI rating.
Remove the 30 amp double pole breaker and install at least one single pole 15 amp breaker. It is best if the other slot is not left open so if you have another single pole breaker of any size install it in the 2nd spot so it isn't open when you replace the cover on the panel. For your 15 amp circuit you need #14 gauge copper wire or larger with a hot, neutral, and ground. Your 30 amp circuit was probably wired with #10 gauge conductors. These will probably work on your 15 amp circuit but it may be difficult to connect #10 conductors to devices.
A circuit breaker protects the wires that the devices are connected to. If the devices that are connected to the circuit are 20 amps the wire size should be #12 wire fed from a 20 amp breaker. This breaker should not trip unless the circuit is overloaded or a fault occurs on the circuit. If the devices that are connected to the circuit are15 amps the wire size should be #14 wire fed from a15 amp breaker. This breaker should not trip unless the circuit is overloaded or a fault occurs on the circuit. Putting 20 amp sockets on this 15 amp circuit will work but the circuit is limited to the amount of load that can be plugged in. You will not get the full capacity of the 20 socket because the breaker will trip at 15 amps.
A 15 amp breaker is not sufficient for a 225 amp load. You would need a breaker that matches the load current, such as a 225 amp breaker. Using a 15 amp breaker for a 225 amp load would cause the breaker to trip immediately due to overcurrent.
It is not recommended to have a resistive load of 12.5 amp on a 15 amp breaker. The general rule is to not load a circuit to more than 80% of its capacity, which in this case would be 12 amps for a 15 amp circuit. Overloading a circuit can lead to overheating and potentially cause a fire hazard.
A 15 amp circuit breaker should trip at 15 amps regardless of the load voltages or impedances. If you have 277 volts and 7 ohms, the current would be 39.5 amps and a 15 amp circuit breaker should trip.
Usually a 15 or 20 amp breaker is sufficient