Learn Russian: Где лежит наш новый договорчик?

This is a common example of corporate or workplace diminutives. Adding the -чик suffix doesn't mean the contract is physically small. Instead, it serves a pragmatic purpose: it softens the question, making the speaker sound more casual, friendly, and less demanding when asking a colleague for a document.

Is it appropriate to use words like договорчик in all business situations?

No, it is highly colloquial. While you will hear it frequently in casual office chatter to make requests feel less bossy, it is often considered unprofessional in strict settings. You should stick to the standard договор in official emails, negotiations, or when speaking with clients and superiors.

Why does the sentence use the verb лежит (lies) instead of just asking where the contract is?

While you can simply ask Где наш новый договорчик?, Russian speakers naturally prefer verbs of position (to stand, to lie, to hang) to describe location. Since a contract is a flat paper document that is typically placed horizontally on a desk or inside a folder, it 'lies' (лежит).

Why is наш новый договорчик in the nominative case if we are looking for it?

Because the contract is the grammatical subject of the sentence. Even though you are the one looking for it, the Russian sentence is literally structured as 'Where is the contract lying?'. Since the contract is doing the action of lying, it remains in the nominative case.