Lines are checked simply by pulling each line at the sail end with a force of approximately 25kg. It is generally thought that lines should easily pass such a test, with the possible exception of Kevlar micro lines. This is a very approximate test, but the best there is at the moment. Line shrinkage is also covered by the same test. A force of 25kg should be sufficient to return a line to close to it's original length, although this does not mean that they need not be checked.
Air permeability is reasonably straight forward, assuming access to suitable equipment.
Strength testing again relies on a much cruder system of manually trying to rip the paraglider material with your hands.