A short certificate is usually the local probate court's certification that an estate proceeding is on record, and most courts will issue them only after the probate is complete, so
First, the easier news: As executor, you should not be legally responsible for paying your father's debts unless, for example, you co-signed on a loan or took some other specific action that would make you legally liable.
What usually happens to a person's debt at death is that all creditors are notified -- and the outstanding debts are ranked in a hierarchy set out in state law and paid off from remaining estate property.
If there isn't enough property to satisfy the creditors at the tail end of the list, they are usually simply out of luck and out of pocket. Highest on the list are the predictably peskiest creditors, such as those demanding payment for expenses of a funeral and last illness, lawyers' fees, and the IRS.
Now the harder news: It may have been easier for you to get the final expenses covered out of your father's estate if you hadn't paid them upfront. But it understandable, and even admirable, that you paid them. Some funeral directors will even insist that grieving and vulnerable family members pay on the spot. And some will sneak some fine print into the bills that obligate family members to be responsible for paying final expenses, so be sure to doublecheck any paperwork you signed.
The probate court that issued the short certificate should have a simple procedure it requires for putting priority on the debts to be paid. If you can present copies of the bills and receipts for the funeral expenses, there is a good chance you will be reimbursed before the other creditors.
the paperwork you were issued here is a little confusing.