I'm no sifu either but yes, you can be allergic to certain mascara/eyeliner. I once tried on this mascara given by my sister and it stings as well. I've used lots of other mascaras and eyeliners but there were all okay. So you might want to try another brand of eye make up. Fake eyelashes won't sting either, you just need a good glue to make sure it doesn't hurt your eyes and it stays.

As for oily face after applying BB cream, I was told that if whatever cream we apply has SPF in it, it's usually more oily. What BB cream are you using?