Learn Russian: По подписании договора секретарь должен выдать вам копию. - Upon signing the agreement, the secretary must issue you a copy.

You are right that по normally takes the Dative case for meanings like along or according to. However, this sentence uses a formal construction where по takes the Prepositional case to mean upon or immediately after. That is why we use по подписании for upon signing.

Since it is in the Prepositional case, why does подписание end in -ии rather than the standard -е?

The nominative form is подписание, which ends in -ие. In Russian, neuter nouns ending in -ие always take -ии in the Prepositional case rather than the standard -е.

Why is договора in the Genitive case here?

Because подписание is a verbal noun meaning the signing. Just like in English where we say the signing of the agreement, Russian uses the Genitive case договора to link the two nouns.

Why use the perfective verb выдать instead of the imperfective выдавать?

The secretary needs to perform a single, completed action in a sequence: first the signing happens, and then they issue the document. The perfective выдать focuses on the successful completion and result of handing you that specific copy.