That's a pretty big fault there and I doubt that static electricity could have caused that.
I recommend that you check the whole circuit for any other strange problems, just in case the cause isn't static electricity.
As for the broken trace, you can always fix it with a wire. Try soldering a wire between damaged ends of the trace and that should fix it. It may be easier to solder the wire on the other side directly between two connectors, since there is very little space left between trace breakage and the connector.
If you decide to work on the trace side, you'll need to carefully scratch off the coating on the trace without damaging the trace itself first. You might be able to solder through the coating, but in my opinion it would be best to remove in on the trace. Later you can cover the joints with insulation tape or buy conformal coating spray and carefully apply it over joints.
If you decide to work on the other side, you'll need to be careful not to use too much heat so as not to damage the connectors. The joints on the other side may be covered by some sort of coating, so test them out with a multimeter first. If you don't get continuity, try scratching or filing the coating off. You may be able to solder through it, but there's no guarantee that you will and you need to be extra careful not to damage the plastic connectors.
As for wire size, there isn't a clear reference there, but I guess that something like standard
\$0.75\mbox{ } mm^2\$ would be strong enough for the current. To be 100% safe, measure the size of the wires coming to the connector on the left and use such wire or larger to fix the open circuit.
If your flight controller already have its own DC/DC converter you should go for it, because it is probably designed for this particular use.
Most speed controllers with BEC are based on linear regulators. Basically these regulators turn excess voltage into heat, the more current you will draw the hotter they will be. If, for some reason, they become too hot they will go into protection mode, and stop to work.
To disable a speed controller BEC just disconnect the red wire from the from servo plug, unless your flight controller board already have a system allowing to choose the power source.
Best Answer
Yes, this is fine, provided you don't draw current in excess of the normal balancing current that the balance wires and connector are rated for. That shouldn't be a problem if it's just running logic and radios, might be more of an issue if you were wanting to run some high-powered servos.
Note that it it's important to only draw from the outer balance wires which are directly connected to the overall pack + and -. If you were to tap off intermediate wires (to get a lower voltage) you'd quickly unbalance the pack.