My questions involve background checks for potential new employees, fingerprinting, developing policies, procedures, and best practices.
Do background checks, fingerprinting, etc., need to be done for all positions? Does it need to be posted in the job advertisement that there will be a background check for the successful candidate or all finalist applicants? Can the background check need to include a financial check and a legal check?
And tangentially, am I correct in my assumption library staff are not considered mandated reporters? Are there guidelines for this as well.
This...is a big question. It's only three short paragraphs. But it's BIG.
It's "BIG" because the risks of getting this topic wrong are immense--from not only the obvious risks involving legal concerns, but risks involving ethics, privacy, and the goal at the heart of the issue: safety.
It's also BIG because the phrase "background check" is not tied to a precise or static definition. When someone says "background check" in the context of employment, here are just a few of the things it could mean:
Each of these "checks" comes with a wide array of legal requirements--or typical legal cautions--governing its use.
For all types of checks, the institution using them should have a clear policy governing what jobs require them, and how such records are evaluated, maintained, and disposed of.
And finally: when developing, implementing, and routinely using any type of background check policy, an organization is wise to take care that it is not incorporating factors that can be shown to disproportionately negatively impact (i.e., discriminate against) a particular category of applicant.
Okay, with all that off my chest, let's answer the actual questions.
Do background checks, fingerprinting, etc., need to be done for all positions?
The degree to which background checks and documentation of identity must be performed are governed by two things: what is legally required, and what the risk management practices of an institution dictate.
These two factors mean that practices will vary from place-to-place. A librarian working within a public school district in the state of New York will be subject to a criminal background check and must be fingerprinted just as any other regular employee within their district. A librarian at a public or association library is not required by law to have a criminal background check, nor to be fingerprinted, but an institution could decide, for risk management purposes, that a position requires that level of scrutiny for safety and security.
Does it need to be posted in the job advertisement that there will be a background check for the successful candidate or all finalist applicants?
There is no requirement in the law that a job advertisement has to disclose a background check in the job advertisement. However, prior to obtaining and using any information from a third party whose business it is to provide background information, an employer must notify an applicant; this notice must be in writing and in a stand-alone format. Further, before a negative decision is made based on such information, it must be disclosed to the applicant. A good resource on this is the Federal Trade Commission, but the third party provider, if they are a true professional, will provide the forms for each of these steps.
Now all that being said, it may be that some local hiring procedures or collective bargaining agreements require the disclosure of background checks in a job notice. Further, some employers may want to disclose their intent to use a background check to avoid surprising candidates further into the process. There is no bar to making such an early disclosure, but if given, such notices should be carefully drafted to avoid implying that those with arrests or criminal convictions will not be considered for the position.
Can the background check need to include a financial check and a legal check?
Yes, absolutely. A background check can include a credit check, a search for liens and other debt instruments, a review of criminal history, a consideration of driving record, and any combination of the items I listed at the top of this reply. Just be careful: if your library or system relies on a third party to supply that information, it must follow the guidance from the Federal Trade Commission (see that link in footnote 6).
Okay, at this point, I have to re-emphasize: before using any type of check, a library should have a policy covering that type of check, and that policy should cover all check-specific legal compliance, as well as: when the check is conducted, how it is conducted, how the information is used, and how the documents related to it are disposed of/retained. 
When developing such a policy, a good rule of thumb for an institution considering any type of background check is to be able to clearly answer the question: "Why are we doing this check?" While the reasons will vary, the answer should always relate to the essential functions listed in the job description, and the nature of your library.
For instance: if a position will create opportunities for a person to spend unsupervised time with vulnerable populations, a criminal background check and rigorous prior employer check is wise. If a position requires a particular credential, verification of that credential makes sense. And if you are hiring someone who will frequently have to drive the bookmobile, a motor vehicle records check is almost always imperative.
On the flip side: if a person is being hired for a job that doesn't require driving, a "current driver's license" should not be required. If a person will never have access to financial information or fiscal resources, a credit check is likely not necessary. And if a would-be library clerk has a DWI that is 20 years old--and no other criminal history--it is likely the conviction is not a basis to eliminate them from consideration.
Last question (and it's another biggie):
And tangentially, am I correct in my assumption library staff are not considered mandated reporters? Are there guidelines for this as well?
"Mandated reporters" is a legal term under Section 413 of the NY Social Services Law. Professionals listed in that section are required to make a report when they:
"...have reasonable cause to suspect that a child coming before them in their professional or official capacity is an abused or maltreated child, [OR] when they have reasonable cause to suspect that a child is an abused or maltreated child where the parent, guardian, custodian or other person legally responsible for such child comes before them in their professional or official capacity and states from personal knowledge facts, conditions or circumstances which, if correct, would render the child an abused or maltreated child."
I have placed a list of the "Mandated Reporters" set by Section 413 below this answer. As you can see by reviewing the (long) list, library employees (unless their function also fits into one of the categories listed in 413) are NOT Mandated Reporters.
Of course, a library--or an institution that hosts a library--can decide and enforce via policy that its employees have an affirmative duty to report observed or suspected child abuse (or any abuse) that occurs on their property or in their programs. Many insurance carriers actually require their insureds to have such a policy.
[NOTE: If an employer has any type of "report abuse" policy, employees should be trained on how to make such reports no less than annually. The average person can have a trauma response to witnessing abuse, which can impact their ability to report it, as well as negatively affect their well-being. Routine training on how to recognize and report concerns, and experienced support for reporters, can help with this.]
Thank you for an important series of questions.
List of "Mandated Reporters" under Section 413 of the Social Services Law (also called "human services professionals"):
...any physician; registered physician assistant; surgeon; medical examiner; coroner; dentist; dental hygienist; osteopath; optometrist; chiropractor; podiatrist; resident; intern; psychologist; registered nurse; social worker; emergency medical technician; licensed creative arts therapist; licensed marriage and family therapist; licensed mental health counselor; licensed psychoanalyst; licensed behavior analyst; certified behavior analyst assistant; hospital personnel engaged in the admission, examination, care or treatment of persons; a Christian Science practitioner; school official, which includes but is not limited to school teacher, school guidance counselor, school psychologist, school social worker, school nurse, school administrator or other school personnel required to hold a teaching or administrative license or certificate; full or part-time compensated school employee required to hold a temporary coaching license or professional coaching certificate; social services worker; employee of a publicly-funded emergency shelter for families with children; director of a children’s overnight camp, summer day camp or traveling summer day camp, as such camps are defined in section thirteen hundred ninety-two of the public health law; day care center worker; school-age child care worker; provider of family or group family day care; employee or volunteer in a residential care facility for children that is licensed, certified or operated by the office of children and family services; or any other child care or foster care worker; mental health professional; substance abuse counselor; alcoholism counselor; all persons credentialed by the office of alcoholism and substance abuse services; employees, who are expected to have regular and substantial contact with children, of a health home or health home care management agency contracting with a health home as designated by the department of health and authorized under section three hundred sixty-five-l of this chapter or such employees who provide home and community based services under a demonstration program pursuant to section eleven hundred fifteen of the federal social security act who are expected to have regular and substantial contact with children; peace officer; police officer; district attorney or assistant district attorney; investigator employed in the office of a district attorney; or other law enforcement official.
 This is why the phrase "Must have no criminal history" or the like must not be included on a job notice. For more information on this, visit https://dhr.ny.gov/protections-people-arrest-and-conviction-records.
 More info on this further into the answer.
 For some employers, this criteria is set by the provider of the organizations’ automobile and/or general liability insurance; this is especially true for organizations that use "company" vehicles.
 Unless there is a very obscure local law I have been unable to find. If you are aware of one, please email me at firstname.lastname@example.org.
 More information on how/when to give this notice is here: https://www.ftc.gov/tips-advice/business-center/guidance/background-checks-what-employers-need-know.
 Or other categories protected by law.
 That's right: I put that in italics, bold, and underlined it! An "Ask the Lawyer" first. No organization should ever "wing" a background check--of any kind. There is too much at stake.
 I know, there is a lot of room for interpretation in this language; when in doubt, seek guidance.
 I think of this as the "Penn State Victims Requirement."
 18 NYCRR § 433.2