If you request a Social Security number (SSN) in item 14, you should receive it a few weeks after you receive the EAD.
If you did not request a SSN on Form I-765, after receiving your EAD, you will need to apply for a Social Security number by going in person to the Social Security Administration. Take your passport; most recent admission stamp in the passport and printout of the electronic I-94 admission record OR photocopies of paper Form I-94 (front and back), if applicable; Form DS-2019, and EAD with you to show work authorization.
Earnings of J-2 visa holders are subject to U.S. Social Security taxes, as well as federal and state income taxes. See IRS Publication 519, "United States Tax Guide for Aliens," available from the IRS website.