If my weapon fails the test after a touch, does the touch get annulled? And what happens if someone gets hurt in the middle of a bout?
Sometimes. Under USA Fencing rules, which follow the FIE, a failed test can annul only the last touch before the fault was found, and only if the fault put the fencer who was hit at a disadvantage. The test has to happen right away, under the referee's supervision, with nothing changed. If you unplug, swap weapons or fiddle with your tip before the referee decides, you lose the right to annulment, and once fencing restarts it is too late to claim. It does not matter where the fault is: weapon, body cord, reel, floor cord or box. It only has to show up clearly at least once during testing.
Annulment is the remedy, never a free point, since referees can almost never award a touch the machine did not register. In foil and sabre, a touch is annulled if a hit by the fencer who was touched failed to register, or if a light against them can be produced without a real hit. In épée, it covers a hit on the guard or strip lighting up, or a good hit not registering. Two épée exceptions: a body cord that simply came unplugged is not grounds, and in a double with one clearly good touch and one doubtful, the fencer with the good touch chooses to keep the double or annul it. If the blade of the fencer who was hit breaks, that touch is annulled unless the break clearly came afterward.
When equipment breaks during a bout, there is no penalty, and touches scored before it broke still count. You are expected to switch to your spare right away, which is why the rules require you to arrive on the strip with two weapons, two body cords and, in foil and sabre, two mask cords, all working. Showing up without spares, or with gear that does not work, is a yellow card, and so is stretching out the interruption. Repairs happen at the armory between bouts, not on the strip, so keep your spares tested and in reach.
For an injury or medical problem that comes up during a bout, current USA Fencing rules allow a break of no more than five minutes once the medical professional on duty confirms it, timed from their assessment and used only for treatment. If they decide you cannot continue, you withdraw. You get another break that day only for a different injury, and calling a medical break the medic does not support is a red card. In direct elimination, withdrawing means your opponent wins the bout, but you keep your place in the results. How unfinished pool bouts are handled is less simple, so ask the referee or bout committee, and check the official rulebook if details matter to you.