No a pap will not show if you can become pregnant or not. There are specific test that must be done to help determine if you have children or not, a pap smear will just test you for cervical cancer and depending which test the MD uses they may also use it to test you for Gonorrhea and Chlamydia
If you want to get on birth control though they will do a pregnancy test to make sure that you are not already pregnant
Yes and no. It tests for certain diseases and those diseases can cause infertility however there really isn't any test besides getting pregnant to know if you can or cannot get pregnant.