Plot hole: Laura rents a nice, large house, furnished with everything she needs, using a fake name, no job, no financial history, has no current bank account or credit cards or any references. She merely hands cash to the agent. She later lands a job without proof of identity, citizenship or residency (legally required), and has no references, no credible work history, or a SSN#. Laura is resourceful, but would lack the means to obtain a convincing fake identity and other false documentation.
Suggested correction: You don't have to be a citizen to work.
You do need an ID, but you don't need to be a citizen.
You need to be a U.S. citizen, a "legal" resident, or have a specific type of work visa to be legally employed in this country. You also need an identity for renting a house or apartment and a prospective tenant doesn't just hand over a large sum of cash for the rent/damage deposit without immediately getting a receipt, while standing on the porch, and not first filling out paperwork.
Plot hole: It makes no sense for the woman from the YWCA to call Martin at his job, which ends up hinting to him that Laura may have faked her death. First of all, how did she get his work number? And Martin never knew anything at all about Laura being at the YWCA, so she was obviously smart enough to NEVER discuss her husband or personal life, such as where she lived, and other personal info. So it seems nonsensical that someone who knew Laura at the YWCA would be calling Martin to offer condolences.
