Менеджер ставит подпись и печать на договор.

The imperfective ставит describes a general or ongoing action (he is in the process of putting his signature and seal). If you wanted to say he has completed the action, you’d use the perfective поставил:
Менеджер поставил подпись и печать на договор.

Both mean “to sign a contract,” but with a slight nuance:
• подписать договор is the standard verb “to sign.”
• ставить подпись на договор literally emphasizes the act of placing the signature—often used when talking about the formal step of adding or stamping. In practice they’re interchangeable in a business context.

• ставить подпись is a fixed verb–noun collocation: you just say “ставить подпись.”
• But when you place something “on” an object (the contract), Russian needs на + accusative to show that spatial relation:
ставить печать на договор (“to put the seal onto the contract”).

• подпись and печать are direct objects of ставить, so they’re in the accusative. For these feminine nouns the form is identical to the nominative, so you don’t see a change.
• договор is the object of the preposition на, which here takes the accusative (indicating direction onto). Masculine inanimate accusative also looks like the nominative: договор.

Yes. Russian word order is flexible for emphasis. All of these are correct:
• Менеджер ставит на договор подпись и печать. (neutral)
• На договор менеджер ставит подпись и печать. (emphasizes the contract)
• Менеджер на договоре ставит подпись и печать. (using locative на договоре shifts nuance: “on the contract” as a location, but less common in legal language)

• Печать here means an official seal or stamp that organizations use to certify documents.
• Штамп can also mean a stamp (e.g., a rubber stamp), but печать is the standard term for the corporate seal in legal/business settings.