The main question is: do they check your visa when you travel (by plane) within the Schengen zone? And the background info of my situation is below:

I'm a non-EU citizen and currently I live in Spain on student visa. Recently I lost my passport which has my Spanish visa in it. I've applied for a new passport, but I cannot get a new Spanish visa unless I go back to NYC and have the consulate there stamp the visa on my new passport. However, the consulate said they would send me this paper that verifies that I'm living legally in Spain. The problem is I have plans to travel to Belgium, Italy and Greece soon and I'm worried that without a visa, I might not be able to either enter those countries or come back to Spain. I know there's no passport control when you get off the plane, but I remember last time I flew from Lisbon to Paris, the check-in guy looked for my visa in the passport. So I don't know if the rules require visa checking before boarding the plan and if I will get in trouble if I only have the my passport and the paper from the consulate and not an official visa.

PS: just to make it clear, I am not a citizen of a country that can enter the Schengen zone without a visa, even for less than 90 days.