Do Banks have a certain amount of discretion in determining when to waive CD early withdrawal penalties or does the Bank have to follow the regulation exactly?

For example, can a bank waive the penalty if a customer needs to cash the CD to have funds to go to a funeral?

Once the CD is past the 6th day (or past 6 days following a partial withdrawal), there is no federally required penalty. It is a contract & customer service issue at that point.

Funeral only works with the owner of the CD.
However, as long as the CD has been on deposit for seven days you can waive penalty.

See Reg D under definitions:

c)(1) Time deposit means:

(i) A deposit that the depositor does not have a right and is not permitted to make withdrawals from within six days after the date of deposit unless the deposit is subject to an early withdrawal penalty of at least seven days' simple interest on amounts withdrawn within the first six days after deposit.1 A time deposit from which partial early withdrawals are permitted must impose additional early withdrawal penalties of at least seven days' simple interest on amounts withdrawn within six days after each partial withdrawal. If such additional early withdrawal penalties are not imposed, the account ceases to be a time deposit. The account may become a savings deposit if it meets the requirements for a saving deposit; otherwise it becomes a transaction account. Time deposit includes funds—

Waiving the regulatory (within 6 day) penalty is discretionary, not required. Although I cannot imagine a bank refusing to waive a penalty in that time period in case of an owner's death, the waiver is not mandatory. Used to be, but no longer.

A business of which I was recently appointed the treasurer has a CD with a bank that told us if we withdraw the funds for operational purposes, we won't be charged a penalty. Just to make sure I understand this correctly, withdrawing funds without penalty is legal and has no ramifications on us or the bank?

After 7 days, early withdrawal penalties on CDs are totally up to the bank and your CD contract.

So if the contract states there will be a penalty and it's waived, that is again up to the bank?

Yup