Under a franchise arrangement, a person or company purchases the ability to run a local business under a larger company's brand. Depending on the franchise arrangement, the local party may have to meet certain standards established by the larger company and purchase products from the larger company as well.
Half-cock is when the position of the hammer is partially but not completely cocked.
They're vastly different from a bluntly-functioning painkiller, for instance. Your hypothesis as to what a half-life means is correct. If the drug has a half-life of 4 hours, you'll still feel 2mg of a 4mg dose 4 hours after the drug has kicked in. 4 hours after that, 1mg is still present.