To answer your question, landlords decide the standards they use to determine who they will or won't rent their apartments to. These standards vary from one landlord to the next. For example, some landlords require an extensive background check, while others only require a deposit. And there is everything in between.
Here, you mention that the applicant has two felonies, but both are non-violent and non-drug related. It is important to understand that mental health problems won't show up in a background search, only criminal activity will (assuming that it has not been expunged). Landlords who do conduct criminal background searches usually are looking for things such as drug charges or violent offenses. A mental health problem, by itself, is not something that a landlord would necessarily be able to screen for; moreover, if the condition is disabling, then the landlord may not be able to discriminate against the applicant on that basis alone without risk of lawsuit.
You also mention that she was granted a temporary restraining order against her ex-boyfriend. The landlord should respect the order, but frankly it is not his responsibility. Even if the landlord gave the ex permission to do work on her place, she has every right to call the police and have the ex arrested for violating the order. The ex has to know that if he goes there he is in violation and subject to arrest and prosecution.
However, if he has not yet been served, then the order is not effective against him yet. Without knowing all the particulars, it is impossible to say whether or not she should have him served at this time or find some place else to stay. That is a decision that only she can make based on her circumstances, her knowledge of him and their relationship - whether she can move, whether it is cost efficient to do so, whether he will react violently, whether he will obey the order and leave her alone, etc etc etc. Only she knows the answers to these questions. She must consider what SHE wants to do based on that knowledge and act accordingly, doing what is best for her and what will make her feel safe.
Lastly, you asked about her getting a new apartment. Unfortunately, there is nothing to make a landlord rent to her. However, if she is low income, then she may qualify for public housing/public housing assistance. I would recommend that she contact her local HUD representative for housing assistance. And, contact her local domestic abuse
center for help with services for victims.