Can I extend #12 wire on a 15A circuit with #14 wire?
You can extend the circuit but it is bad practice to put #14 awg wire on a circuit that originates with #12 awg wire. For example; the reason is because someone tosses a mini fridge on the circuit and now the breaker trips, going back to the panel you many years later or the next owner sees #12 wire and installs a 20 amp breaker not knowing or remembering there was 14 in the run. This can result in a fire especially with back stabs in use and the smaller wire size. I have actually found this exact change of events on several occasions over the years. 12 awg is not that much more than 14 awg so it is best to continue with 12 awg , if you go cheap make sure to label the wiring in the service panel as connected to #14 , but even that is a bad practice compared to running #12.