When will the visa be checked when leaving the country?

The visa will be checked when leaving the country and arriving at the destination. When you go abroad, people from the Exit-Entry Administration will check your passport and visa; after arriving at your destination, the immigration bureau of the destination country will check your passport and visa.

You need to apply for a passport first, and it will probably take more than a week to get it.

You cannot apply for a visa until you get your passport, because if your visa is approved, the embassy will print and paste the visa directly on your passport.