Re: How do spiders breathe inside of their eggs (i.e.

Insects have tiny pores (one to several microns wide) in their eggshells 
that allow atmospheric oxygen to cross into the inside of the egg.  These 
are called aeropyles.   I assume (although I have not found much in the 
way of literature on the subject in my short search) that spider eggs are 
similar.  In other words terrestrial arthropod eggs are permeable to 
oxygen.

Reference:

Hinton, H. E. 1970. Insect Eggshells.   Scientific American.