Learn Russian: Компания оплатила счета по окончании проекта. - The company paid the bills upon completion of the project.

This is a great example of semantic differentiation in plurals. Счета (with a stressed -а) means bills, invoices, or bank accounts. The other plural form, счёты, means an abacus or personal scores (like in the phrase сводить счёты - to settle scores). Since the company is paying, we need the financial meaning: счета.

Why does окончание end in -ии here instead of taking the Dative case after по?

When the preposition по means upon or immediately after an event, it takes the Prepositional case, not its usual Dative. Because окончание (completion) is a neuter noun ending in -ие, its Prepositional ending is -ии. Thus, по окончании means upon completion.

Why is it оплатила счета without the preposition за?

The verb оплатить (to pay for or to cover the cost of) is transitive, meaning it directly takes an Accusative object without a preposition. You would only use за + Accusative if you were using the related verb заплатить (for example, заплатила за счета).

Does компания mean both a business company and a campaign?