>>2942551
>1A should be plenty for a 12-inch strip with 12 LEDs
Yes that is likely the case. 12V 1A is 12W, spread across 12 LEDs is 1W per LED, usually those LEDs handle half a watt each at the maimxum. You can measure the dimensions of your LEDs to check what package they are to know for sure, pic related. You can also usually calculate 12V LED strip current draw from the values on the resistors, by assuming each resistor is dropping 3V.

When it comes to short distances of wire, the limiting factor is how much the wire heats up when the current of the load runs through it. Thicker wire heats up less because it resists less, P = R*(I^2). So long as your wire is all rated for an ampacity of 1A or more, it doesn't matter if you go to and from 12awg or 18awg or whatever.

The 12V 1A wall brick is going to be your main insurance against a house fire. Modern power bricks like that turn off their output when you draw too much current from them, so there shouldn't be any risk of fault even if there's a short-circuit at the output. That said, it depends on how much you trust your power brick, so you may consider putting a fuse in series with the power supply, as close to the power source as is reasonable. Over-rating the power supply (e.g. using a 1A supply on a 0.2A load) is also going to result in the power supply remaining cooler when operating, which is probably the case for you anyhow.