It is theorized that IF spontaneous combustion is real that static electricity may spark it off. instances are so rare though that it may all be an urban myth.

If it is built up somewhere (10th grade shop, teacher nearly lost it, we placed the Vandegraph generator against a steel wall), and some years ago a sailor onboard a ship died after placing a nine volt meter against his heart area (he had an unknown, until then, defect).

Well technically, a lightning bolt is static electricity. So yes, you can die.
But a typical lightning bolt contains 1 billion volts and contains between 10,000 to 200,000 amperes of current.
so other than getting hit by one, there is no problem.

Even a Van de Graaff generator, which will give somewhere from 10,000 to 30,000 volts, but at a very low amperage, will only give a little spark, or when charged will make your hair raise.
So the static one could generate from just around the house is less than 100 times than that.

But as far as I've ever learned, it's just not enough to be harmful at all.

The main factor isn't the Volts, it's the Amps, and it's not possible to generate enough amps to harm you in normal day to day situations.
Your cat is fine.

No.

Try inflating some balloons and sticking them to the cat so that he/she will not get injured when making sudden moves after being shocked.

Tell your cat to wear a chain around the neck, just long enough to stay grounded at all times, then he will be fine ????