Learn Hardcore Russian: Менеджер ставит подпись и печать на договор. - The manager puts a signature and a seal on the contract.

Literally, it says “The manager puts (places) a signature and a seal on the contract.” In natural English you’d normally say: “The manager signs and stamps the contract.”

Why is the verb ставит in the imperfective present tense instead of a perfective form?

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.

What is the difference between saying подписать договор and ставить подпись 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.

Why is there no preposition before подпись, but there is на before договор?

• 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”).

What case are подпись, печать and договор in?

• 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.

Could you reorder the sentence? For example, put договор before подпись and печать?

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)

What exactly is печать in this context? Is it the same as штамп?

• 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.

Why are there no articles like “a” or “the” in the Russian sentence?

Russian does not have articles. Definiteness or indefiniteness (a/the) is usually understood from context or added with words like этот (this) or какой-то (some).

Could you replace ставит with печатáет (from the verb печатать)?

No, that would change the meaning to “prints” in the sense of reproducing text. If you want to say “he stamps” you could use штампует, but in business language ставить печать (to place the seal) is far more common.