You should take all of your paperwork, from the US and Canadian immigration cases and go see an experienced US immigration attorney. You may have unexplored options.
The answer provided here is general in nature and does not take into account other factors that may need to be reviewed for a more precise answer. You should consult with an immigration attorney before taking any action. The answer here is not intended to create an attorney-client relationship.
Nothing. Unless she is eligible for EAD, there will be no Social Security Number.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: firstname.lastname@example.org; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.
Not all asylum applicants are eligible for a work permit, which is the usual way to obtain a social security number. An attorney can review your asylum case to see if you or your daughter would be eligible for one. Sometimes there is an easy way and sometimes a hard way and sometimes it is not possible to obtain a work permit while an asylum application is pending. There may be other options for your daughter, and for you, so you should not delay in speaking to an experienced immigration law attorney.
The answer above is only general in nature and cannot be construed as legal advice, given that not enough facts are known and detailed research has not been undertaken. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers require an investigation into all facts about your case. Any answers offered on Avvo are of a general nature only and are not meant to create an attorney-client relationship. Use these answers at your own risk.