Would an electric fence be able to detect being bypassed?

The only way a detection could be feasible is looking for changes in signal reflection. Neither resistance nor inductance will provide reliable information of bypassing activities, especially not with a mile long fence wire. However, with a sharp peaking signal (already present due to the nature of the electric fence) and analysis of the electrical reflection pattern might provide an usable information to detect bypassing. The term you want to look for is Time Domain Reflectometry. See: https://en.wikipedia.org/wiki/Time-domain_reflectometry

Theoretically, yes. You can measure the amount of resistance and determine the length of the fence.

However, there are some practical limitations to this. Doing this accurately would require quite an investment while the resistance won't be stable. Even the weather would influence it. Compensating for all those fluctuations would require some pretty advanced equipment while an electric fence is essentially a very low-tech device.

are electric fences constantly charged?

No, they are usually pulsed. This has a couple of advantages, one of them making it easier to generate such a high voltage. They are continuously powered, but the circuit is not designed to sustain the charge.

As @PlasmaHH mentions, if you want to measure your fence the proper way, impedance would be more valuable than resistance. I have never seen anyone do this on an electric fence, but you can measure the length of a coax cable by sending a pulse through it. The time it takes to come back and the form of the pulse will tell you a lot about the characteristics of the cable, including the length. However, this wouldn't be foolproof on an electric fence.

I am not going to show this to my family (we meet monthly at the family farm where there are multiple electric fences), because they would laugh at it. Electric fences are designed for maximum reliability when installed by someone with only the barest understanding of electricity. As such they have no components not needed except for an oversize case to protect it from irate cows. Further the traditional wire of choice has been iron which has high resistance, and rusts which increases resistance in unpredictable ways. These days the wire of choice in nylon carbon fiber mesh which in addition to having impossible to calculate resistance, the impedance and capacitance also are affected by the weather. To build an alarmed electric fence would require not only a entirely new class of "charger" (the box that charges the fence), but different wire, insulators, poles and grounding.