Would it be safe to attach a nichrome coil to copper wire and run electricity through it to make a heating coil?

Obvoiusly I would not electricute or burn myself, I would only leave it on for 20 min at a time and I am thinking about soldering the copper wire to the coil, which would be in a box.

You know, that's the way the pros do it.  If you've ever looked at the  insides of a toaster or hairdrier, well,  it is just as you say:  nichrome wire is attached to copper wire, and an electric current runs through both of them. 

Explaining why the nichrome wire gets hot, but the copper wire does not, is left as mental exercise for the interested reader.

There are some other things to think about too, like the voltage across the heating coil, and the current that flows through it.  The product of those two numbers is the power dissipated as heat.  Also you can model your length of nichrome wire as a resistor, R, and use Ohm's law to answer those other questions.  There's a small temperature dependence too.  The total resistance of the element might be as high as 10% greater when hot.  See:

As others have mentioned, soldering to nichrome is a bad idea because the solder can melt when the nichrome gets hot. 

Regarding safety:  your safety is in your hands.  I recommend not touching coils of nichrome wire while they are hot, or electrically energized.  Safety glasses are probably a good idea too.
You may want to use a screw terminal or some other physical connection. The heated wire will likely melt the solder you use.
Assuming the joints are sound, yes, its what you do to make a heating coil....