Fault.

A falls off B's roof while fixing it.

In this situation B is not at fault. In the case that B is to blame, B would have to be responsible for whatever A's reason was to fall, which was most likely the roof, and if B could fix the roof then he would not have asked A to fix it.

A is homeless and begging on the street. B walks by, does nothing and just keeps walking.

Again B is not to blame. A's way of life or lack of money is not B's responsibility. In my opinion the only way B would be responsible for A's life is if B was A's guardian and A is under the age of 18.

B lends his car to A, which has some problems, and A get's in an accident.

The blame can only be placed on whoever first suggested that A uses the car. If A asked for the car then it is A's fault, and vice versa.

B has the piggy virus, goes to school, and gives it to A.

The blame certainly does not lie with A. And the only reason B would be at fault is if B knew that he was ill.