Do some research on capacitive coupling, and Inductive coupling. This should explain why you are seeing voltage on the "dead" traveler.
If you load up the traveler that is not carrying current; with a test light for example, you'll likely see this voltage go away.
Working in a residential setting, this voltage is likely harmless. Phantom voltages, however, can be lethal when working with distribution lines and other high voltage situations.
You may still want to have a licensed electrician come out and take a look, just to be sure this is actually the case.
Yes, it is a safety issue, wiring should always be covered by something, anywhere. The connection should be water tight, the connector used is not. Electric proof is not an issue, that is the job of the insulation of the wires. Cold resistant is not an issue for the wiring, but could be an issue if the cold made some protective material brittle. Protection from weather is an issue, you do not want water or debris entering the wiring space.
I suspect the back fill that the tubing is placed in settled, pulling the buried portion away from the above ground portion. If you can't pull the sections together to reattach, you may have a problem. It may be worth digging up the tubing to get more play to reattach, as the only code compliant fix I know of is to either replace the entire run, reusing materials where possible, or insert a new box to make up for the gap. You'll have to cut the wires to place the box. There may be enough slack to reattach them, otherwise, pigtail them together with short new lengths. Be sure to use water tight box and connectors if you go this route. The box must remain accessible.
I shouldn't even mention this. The hillbilly fix would be to get a length of plastic water tubing, the thin walled gray stuff (PB?) is what I've seen, of adequate diameter. Slit it along it's length, wrap it around the open joint, and seal the overlap and the tubing above and below with silicone sealant. Clamp in place with several screw type hose clamps. Far from code compliant but it adequately protects the wiring. More water tight than the current connector.
Best Answer
The break in the wire can be confirmed by placing a voltage on one end of the pair then measuring for it on the other end.
If you know you have voltage on one end then simply having lack of voltage on the other end is proof positive.
If you have 3 wires to work with, then you can use a 12 Volt battery like from a car, then put it across pairs of wires, using a multimeter on the other end, to determine precisely which wire is broken.
If a trenching machine is responsible then it is most likely they all have been cut.
The precise location of the break can be located by using a time-domain reflectometer.
If the gas company came out and dug and you haven't had power ever since then their denials mean nothing until they come out and prove that they did not cause it.
I wouldn't let them off the hook until they send out technicians to prove their denial.
I would go so far as to tell them you think there is live electricity right near where they dug for the gas pipe. See if that will get them off the dime.