Hi Jerry, first ALL electrical wiring is something to be wary of, however, telephone wiring is very small voltage, and not anything to be concerned with unless you are dealing with a wet environment. As for testing the wiring, use a volt/multi-meter to determine if there is a current/continuitity from one jack to another.
Now, I'll take a wild guess on what the actual problems may be. First, the white deposits could be stopping the current (i.e., just like on a car battery terminal with corrosion), so clean this deposit off with steel wool or sandpaper. Second, quite often outlets will be "daisy-chained" from one to another and if there is a disconnected wire after the good outlet in the chain then every outlet downstream will be dead. To check this use your multi-meter and check for continuity from one to another. If the span from one jack to the next is too long for the connections on the meter, take a piece of wire to make the connection.
Finally, if the household had more than one active line at some point in time, then the second set of wires would of been used and you would have to make some crossovers for the phone signal to travel downstream.