Believe it or not, I actually do this type of testing for a living, so let me assure you - CAT 5 cable will carry 7A of AC current for approximately 2.5 seconds before fusing. 30A will last approximately 65ms before fusing, and 60A will last approximately 13ms before fusing.
I will not state how much current can be 'safely' carried by CAT 5 cable for your own safety...
Forgot to answer the second part of the question...you asked why would 120V cause the CAT5 cable to fuse when 12DC does not:
DC current will use the entire cross-sectional area of the wire to conduct current. I do not know what the DC current rating of CAT5 cable is, but if you say you can carry 7A DC through it, I assume the cross-sectional area of 26 AWG wire is sufficient...
With respect to AC current - as frequency increases, conductors begin to exhibit "skin effect," which is that the current actually travels on the outer surface of the wire and not through the center of the wire. As frequency gets higher, the skin depth gets smaller, so you are effectively reducing the area that the current is flowing though, so at 120Hz, you really don't have the equivalent of 26AWG copper wiring any more - you have something less (I don't know how much less, unfortunately). At extremely high frequencies, the entire current path may be only on the surface of the wire itself. 30 Megabit Ethernet signals are very low voltage and very low current, so the amount of copper needed to transmit the data is very small, so why then is CAT5 cable 26 AWG??
The answer is simple - the telecommunications company that is sending service into your home has engineered their protection mechanisms to ensure that the most dangerous voltages your telecomm lines will ever see is 120V. They also know that if you should get 120V on your telecomm lines, the equipment that your telecomm lines are connected to will have secondary voltage-limiting and secondary current-limiting protection. In short, they know you have sidactors and fuses in your equipment. Fuses are designed to carry up to 20% above their rated spec for up to 15 minutes, so a 2A fuse will carry up to approximately 2.4A for a full 15 minutes before fusing. Anything more than that, and the fuse operates almost immediately. When that fuse operates, the current is interrupted and your telecomm lines don't fuse and catch your house on fire...so, that CAT5 cable has to be rated to carry up to approximately 2.5A for 15 minutes...
Hope this helps. 