My pet keeps chewing cords, and I haven't found a good solution. Will getting surge protectors help keep the pet safe from electrocution if they chew through the coating into the wire?
No, a surge protector will not.
An RCD (Residual Current Device) or GFCI (Ground Fault Circuit Interrupter) aka ELCB (Earth Leakage Circuit Breaker) are the devices that can provide protection, BUT you should be aware that nothing is 100% - these devices can fail...
You should get the cords away from access by your pet, shielding or protective strips or even moving the cables out of reach are possibilities.
TLDR: Make GFCI protection a top priority.
No, a surge protector definitely doesn't do that.
GFCI (called RCD in Europe)
The single biggest help is going to be GFCI protection. How exactly to apply that is beyond the scope of this question, but any receptacle such a cord is plugged into should be GFCI protected.
Current normally flows in a loop (circuit) - out the hot wire and back on the neutral wire. Electricity moves at the speed of light, so current flows should always be equal. The GFCI is called a "ground fault" detector, but what it's actually looking for is differences between current flow on hot and neutral. That indicates current is taking a third path which is unintended, and thus potentially dangerous.
If your pet chews through hot and ground, or neutral and ground, there will be current flow through his mouth and that will trip the GFCI.
However, your pet has a chance of chewing through hot and neutral only. In that case, current will flow between them through his mouth, and since this is normal current flow, the GFCI won't trip, and he will be shocked and burn his mouth.
The good news is the current probably won't go farther into his body. So it probably won't kill him (though you never know with electricity).
AFCI is of limited value
Another type of protection is AFCI, or arc fault protection. This is a digital signal processor inside the circuit breaker that is listening for the crackle-snap sound of arcing (you know, that sound when you hook up a speaker or twist a headphone plug).
An AFCI can't detect the dog getting shocked. It could detect "wires shorting together" type arcing if he chewed the wires enough to get hot-neutral or hot-ground to short to each other. But I seriously doubt he'd get that far.
AFCI will, however, protect you from what comes later when you try to continue using that cord.
Not usually. Surge protection won't help as your problem here is simple current flowing from one wire to another through your pet. Surge protectors typically include a circuit breaker for overcurrent protection, but that will not typically react fast enough to prevent electrocution.
There are, however, two things that can help:
A ground fault circuit interrupter, typically installed either as part of the circuit breaker in the main panel or as a combination duplex receptacle/GFCI. A GFCI watches for a difference between hot & neutral - i.e., leakage to ground, including through a pet or a person. It reacts at a current level and fast enough speed to prevent electrocution. This will not stop all forms of electrocution - if the current literally flows in one part of an animal and out through another part back to the original wires then GFCI won't "know" there is anything wrong. But this is your basic line of defense with electrocution concerns and is a really good idea in your situation if the receptacles are not already protected as required by modern code in kitchens, bathrooms and certain other areas.
An arc fault circuit interrupter, typically installed as part of the circuit breaker though occasionally installed elsewhere in a circuit, protects against certain types of wiring problems that result in arcing, which causes fires. An AFCI does not typically prevent electrocution directly, but the process of chewing through wires can result in damage that would trigger an AFCI. AFCI is now required in most parts of a house that are not protected by GFCI, but older houses generally do not have AFCI installed. Installation of an AFCI circuit breaker, if you don't have one already, would add an additional level of protection, but not a guarantee, for your situation.