Yes you can do this, most laser used for cutting work like this. Something emits light then it is highly focused to heat the material to very high temperatures. An object in general can only get as hot as the thing that is heating up if both are acting like perfect black bodies. This is because when something is heating up it emits light waves back out. If the temperature of the light is increasing the amount of energy in the waves increases as well. This goes till the 2 are balanced out so when the energy of the incoming light is equal to the energy of the outgoing light then you reach a steady state temp.
For most cases no. You can only focus light to reach the temperature of the light source. It does *not* increase to infinity as you focus more light into a smaller area. A light bulb in a flash light is not very hot so you won’t achieve a fire with it. The moon acts as a reflector so the question gets more complicated. Technically the light source is the sun but I do not know how having a reflector between the source and focus point affects the results.
To specifically address your question about the full moon: [no, you can not](https://what-if.xkcd.com/145/).