Disabled for tax purposes means you are unable to perform substantial gainful work.
You are considered disabled if you can furnish proof that you can't do any substantial gainful activity because of your physical or mental condition. A physician must determine that your condition can be expected to result in death or to be of long, continued, and indefinite duration.
https://www.irs.gov/retirement-plans/plan-participant-employee/retirement-topics-disability
It is not required that you be approved as disabled by your company's disability carrier or social security, although those things would certainly help. As noted in the rules, you only need a doctor's certification.
However, there are many people who meet a medical definition of disabled, but who can still perform substantial gainful work--amputees, for example. They might not be able to perform their exact previous job, but if they can perform other gainful work, they aren't disabled. So it depends on your facts and circumstances.
If you claim the disability exception to the penalty, you don't send proof with your tax return, but keep the proof with your other tax records for at least 3 years.
Yes, if your doctor has declared you permanently and totally disabled, you can claim the penalty exemption. You not need to send proof with your return, however, you will need to keep it with your records in case the IRS would contact you for proof of the disability.
The approval from social security does not determine whether or not you are disabled permanently, although it makes it slightly easier to prove. So yes, you can still claim it even without your application being approved.