You are confused on some things specifically filed vs served/prepared.
A five day notice to pay rent or quit needs no official stamp nor to be "filed" with the courthouse before you are served with it. After that notice expires an eviction case is then FILED with the courthouse which results in a summons/complaints served upon you.
That said, many companies use a third party to prepare and serve their pay/quit notices. And any cost assoicated with that can be sought from the tenant.
You say you are trying to pay as quick as you can...you need to pay on or before the due date to avoid this time of action against you. Only if at the time the notice was served you owed NO outstanding rent would you not owe the fee. If a court case was already filed, you would owe the filing fees too.
If you have to pull out receipts and stuff..sounds like you are not paying rent all in one lump sum w/o having to deal with late fees, etc...that is what should be done to avoid this.