It is not illegal per se to test or launch a rocket from your back garden, but in all countries these practices are well regulated. The governing regulations depend on the size and power of the rocket. To give you a flavour of the regs you can click on the following link for a summary of US FAA home rocket regs.
http://www.seanet.com/~ssstolt/regs/faaregs.htm
.In answer your second question, if you were responsible for a rocket hitting a plane (accidently or not) you would be deeply in the mire.