Background check

I have never heard of those two. Just employment status and income. Possibly illegal. Check the new york city 311 website.

Why would it possibly be illegal. Give me your name and I could run a criminal background check on you. And if you can ask employment status, and employment in the US requires the right to be resident in the U.S., then it is certainly allowable to know the status of the potential renter.
There is no obligation for a landlord to have illegal immigrant felons renting from them.

Ironic that it might be illegal to check whether a person had previously broken the law.

Don't think that those fall under fair housing. Persons with a criminal background would not be a protected class. So I believe it is legal. Credit checks for housing now commonly include a search for sex offenders as well.

Criminal background is completely legal. While most ask re: convictions, there is no law I am aware of barring someone applying for housing (e.g., in a coop interview) if they have ever been arrested regardless of conviction, or if they have ever been subject to a restraining order. Immigration status is more dicey. Someone else can weigh in on that.

immigration status more dicey? No, If you have no right to be here you have no right to expect to rent an apartment. But, for argument's sake, assume it is illegal. So, make sure your "rights" are followed. Sue - go to civil court and sue and expect not to be arrested (whether by a requirement by a officer of the court (which would include the clerk) to report illegalities, or by an angry landlord who takes revenge) or to be around long enough to make the court appearance. Or go and bring a complaint to the police, right try that one.
Dumb PC crap.

The politics of this don't concern me on a RE forum. I'm only discussing the legalities. In a coop, if you are on the board and owe the corporation a fiduciary duty of care, then you had best get specific legal counsel if you want to delve into areas of national origin and specifics of immigration status beyond legal and illegal.

In a rental situation, if every illegal question imaginable were asked of you, you would indeed by hard-pressed to actually pursue any courses of redress. Practically speaking, the remedies are cumbersome, slow, and for most people more trouble than the effort is worth. If a landlord asks your marital status and the race of your spouse, it is illegal but where does that leave you? The time and money needed to pursue that breach of the law are more than most people are willing to invest. Particularly when the task at hand remains finding a place to live.

Which is why I said check the 311 website.