If you are in L-2 status, you need an EAD to work. You need to file I-765. In the event you do not receive your EAD within 90 days, you are eligible to receive employment authorization for 240 days. When you file I-765, indicate your current SSN on the form, but no need to send a copy of it to USCIS. You should not apply for new SSN, but check your SS card and make sure it says that "valid for work only with USCIS authorization." (This is what you should have based on your previous L status.) If it says that it is not valid for work, contact the SS office and request a new/revised SS card (not a new SS number).
No, that by itself is not enough. You need to have a valid EAD card issued to you to be able to accept employment.
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No you may not work you would need authorization. I would verify this with social security as it is beyond the area of my expertise but I believe you are issued one SSN and that stays with you.