I have build couple level 3 corporation

I have build couple level 3 corporation. However, my game level drop from 3 to 2 for some months. Will the corporation that need level 3 still produce product during the month I am in level 2?

Yes, the game level only determines what corps you can start.
Once the corps are built, it doesn't matter what your game level is, they'll produce so long as they have the materials and workers they need.