Can one bring a Shir Hamolas card into the labor room to be visible during labor (for the woman to feel safer)? Can it be see through? Would these issues apply to a picture of a Tzadik?

 

It  must be covered with two covers, one of then not visible.

This does not apply according to Halacha to a picture of a Tzadik. Nonetheless, Chassidim have a similar Hergesh with pictures of a Rebbe.

See here for more details.