Yes, AB 1733 is only for issuance of authorized live birth certificates.

Yes.

A homeless person, or a homeless child, or youth on behalf of themselves can make a request for a birth certificate, or any person lawfully entitled to request a birth certificate on behalf of a child, if the child has been verified as a homeless person, or a homeless child, or youth.

Pursuant to Health and Safety Code Section 103577, a "homeless services provider" includes:

“(A) A governmental or nonprofit agency receiving federal, state, or county or municipal   funding to provide services to a “homeless person” or “homeless child or youth,” or that is otherwise sanctioned to provide those services by a local homeless continuum of care organization.
(B) An attorney licensed to practice law in this state.
(C) A local educational agency liaison for homeless children and youth designated as such pursuant to Section 11432(g)(1)(J)(ii) of Title 42 of the United States Code, or a school social worker.
(D) A human services provider or public social services provider funded by the State of California to provide homeless children or youth services, health services, mental or behavioral health services, substance use disorder services, or public assistance or employment services.
(E) A law enforcement officer designated as a liaison to the homeless population by a local police department or sheriff’s department within the state.”

AB 1733 states that providers may not charge a fee for verification of eligibility.

Counties may assist in searching for the record before the applicant makes a request. 
HSC Section 103575 provides that the state or local registrar or county recorder may, without a fee, verify date and place of birth, when the applicant presents sufficient information to identify the birth record.  Counties may also refer applicants to the Quality Assurance and Production (QAP) Section with the State Registrar at (916) 552-8116 to verify a county of birth.  The homeless applicant can then contact the appropriate county to obtain the birth record.

Yes, as long as the person is considered a homeless child or youth.  An authorized third party (i.e., parent, guardian, legal guardian, grandparent, brother, sister) other than the registrant may request a copy of a birth certificate on behalf of a homeless child or youth as long as they are lawfully entitled.

The best way to file an application is to submit in-person at a local county office where a county official may take a statement sworn under penalty of perjury (HSC Section 103526). Alternatively, a mail-in package consisting of an application, affidavit of homelessness, and sworn statement, can be sent to the State Registrar. The State Registrar does not offer in-person service.

A person applying for a certified record of live birth is entitled to only one birth record per application for each eligible person verified as a homeless person or a homeless child or youth.

Yes, the completed request would be mailed back to the address indicated on the application.

Pursuant to 42 United States Code Section 11302, the general definition of a homeless person or homeless child or youth is:

“(a) In general
For purposes of this chapter, the terms “homeless,” “homeless individual,” and “homeless person” means —
(1) an individual or family who lacks a fixed, regular, and adequate nighttime residence;
(2) an individual or family with a primary nighttime residence that is a public or private place not designed for or ordinarily used as a regular sleeping accommodation for human beings, including a car, park, abandoned building, bus or train station, airport, or camping ground;
(3) an individual or family living in a supervised publicly or privately operated shelter designated to provide temporary living arrangements (including hotels and motels paid for by Federal, State, or local government programs for low-income individuals or by charitable organizations, congregate shelters, and transitional housing);
(4) an individual who resided in a shelter or place not meant for human habitation and who is exiting an institution where he or she temporarily resided;
(5) an individual or family who —
(A) will imminently lose their housing, including housing they own, rent, or live in
without paying rent, are sharing with others, and rooms in hotels or motels not
paid for by Federal, State, or local government programs for low-income
individuals or by charitable organizations, as evidenced by —
(i) a court order resulting from an eviction action that notifies the individual or family that they must leave within 14 days;
(ii) the individual or family having a primary nighttime residence that is a
room in a hotel or motel and where they lack the resources necessary to
reside there for more than 14 days; or
(iii) credible evidence indicating that the owner or renter of the housing will not allow the individual or family to stay for more than 14 days, and any oral statement from an individual or family seeking homeless assistance that is found to be credible shall be considered credible evidence for purposes of this clause;
(B) has no subsequent residence identified; and
(C) lacks the resources or support networks needed to obtain other permanent
housing; and
(6) unaccompanied youth and homeless families with children and youth defined as homeless under other Federal statutes who —
(A) have experienced a long term period without living independently in
permanent housing,
(B) have experienced persistent instability as measured by frequent moves over
such period, and
(C) can be expected to continue in such status for an extended period of time
because of chronic disabilities, chronic physical health or mental health
conditions, substance addiction, histories of domestic violence or childhood
abuse, the presence of a child or youth with a disability, or multiple barriers to
employment.”

42 United States Code Section 11434a – Definitions
For purposes of this part:

“(1) The terms “enroll” and “enrollment” include attending classes and participating fully in school activities.
(2) The term “homeless children and youths”—
(A) means individuals who lack a fixed, regular, and adequate nighttime residence (within the meaning of section 11302(a)(1) of this title); and
(B) includes—
(i) children and youths who are sharing the housing of other persons due to loss of housing, economic hardship, or a similar reason; are living in motels, hotels, trailer parks, or camping grounds due to the lack of alternative adequate accommodations; are living in emergency or transitional shelters; are abandoned in hospitals; or are awaiting foster care placement;
(ii) children and youths who have a primary nighttime residence that is a public or private place not designed for or ordinarily used as a regular sleeping accommodation for human beings (within the meaning of section 11302(a)(2)(C) 1 of this title);
(iii) children and youths who are living in cars, parks, public spaces, abandoned buildings, substandard housing, bus or train stations, or similar settings; and
(iv) migratory children (as such term is defined in section 6399 of title 20) who qualify as homeless for the purposes of this part because the children are living in circumstances described in clauses (i) through (iii).
(3) The terms “local educational agency” and “State educational agency” have the meanings given such terms in section 7801 of title 20.
(4) The term “Secretary” means the Secretary of Education.
(5) The term “State” means each of the 50 States, the District of Columbia, and the Commonwealth of Puerto Rico.
(6) The term “unaccompanied youth” includes a youth not in the physical custody of a parent or guardian”.”