This adapter will not work with your pet fence. You can sometimes get away with a slightly higher voltage rating, but the power unit you have only supplies a fraction of the amperage you need, so it won't work.

You should get the power unit recommended by the manufacturer.

No, this would not be a safe solution. You need to cut the cable completely and bring the ends into a junction box to splice it back together properly. Also, the junction box can't be buried in the wall when you're finished - you need to leave it accessible.

As long as the voltage is the same (and it actually fits in your phone) you can vary the amperage. A higher amp-hour (aH) rating means longer run time between charges.

I was told that electrons travel on the surface of copper wire; therefore, stranded wire can carry more electrons than solid core wire because of the increased surface area. If this is true, then why is solid core wire used so much more extensively than stranded?

It depends on the application, and the size of the wire. For instance, you won't find solid wire in sizes greater than 8 commonly available, and the Code specifies that over a certain gauge, wire must be stranded for ease of installation.

Technically, solid wire has a higher degree of conductivity, since there is less resistance inherent in the wire over long runs, where the increased surface area of stranded wire leads to a higher resistance rating of the cable and makes it more prone to overheating at high current loads.