Yes of course, power is power, all 110volt, what ever the source in your house (except a dryer circuit that is usually higher).
The difference is the amperage of the circuit. Lighting circuits have amperage controlling breakers that are usually 15amp, power circuits are usually 20amp but excepting a very very unusual condition you would not need to worry about it.
If you want to you could remove the light socket and put in an outlet then plug your light into the outlet.
no you can't
No, it will cause the bulb to blow. A 12 volt light is DC voltage and your home outlet is AC voltage.
It is likely broken if there is power to the outlet
Plug a lamp into a working socket of an outlet to ensure the lamp works. Then turn off the switch and plug lamp into all outlet sockets, top and bottom outlets until the lamp doesn't light. Then turn on the switch and the lamp should light. Often a switch in a bedroom is only connected to one socket in a duplex outlet. It is possible to remove a jumper in an outlet to isolate the sockets for just this purpose. Often an electrician will install the outlet upside down (The third prong pointing up, to identify the outlet.
plug it into any outlet
You can make music come on with a light switch the same way you can make a light bulb come on - plug a radio into the switched outlet.
Powed cords always plug into an outlet.
Plug a plug in an outlet :)
You plug it in your plug outlet
an AC Adapter allows you to plug your computer into a standard wall outlet
Plug and socket, plug and socket outlet, plug and receptacle or plain plug and outlet all seem to be in common usage in the US. Plug and socket is possibly the only wording commonly used in the UK. [Plug and socket outlet sounds ok to a me, as a Brit, but we never really use that expression in the UK. Plug and outlet or plug and receptacle actually sound very strange!]