Hey, i have a same situation, I have E2 visa , I’m in usa and i won in dv lottery and unfortunately I worked for one day with doordash , so are sure it’s ok if i want to do CP ? Because someone told me he won in dv lottery and he did CP in his country and he was on ji visa before and they asked him if he was do unauthorized work.1. Sounds like your work is not allowed under F1 so you have violated your status and are not eligible to process AOS. Lawyer can’t change the law. You will need to do CP.
2. N/a, but you should stop violating your status anyway.
3. If the DS260 asks if you have ever been issued a ssn then yes you have to answer truthfully.
Note that separarely from this there have been a number of instances (reported on another forum I’m on) where people who have worked in violation of their F1 status, exactly in this kind of job, have been refused re-entry on their F1s and in some cases have been handed a 5-year bar to the US. There is currently just about zero tolerance if you are discovered breaking visa rules. The longer you do it, the more you risk your status. Violation of status is not generally a problem for consular processing as such, but if you are discovered in the interim and handed a bar on entry then that cannot be overcome for DV.