Assalaamu alaykum wa rahmatullaah. There is a woman who likes a man and wants to get to know him for the purpose of marriage (they do not have a relationship). The woman found out from the man's mother that he is religious and used to pray, knows who Allah is, and does not drink etc., but he does not pray at the moment. The woman's intention is to not marry him if he does not start to pray, but she wants to see if there is any chance that he may start to pray and thinks that there is some potantiel. Her intention is to set that condition. Now, is it permissble to let this man know through a third person that she is interested in him and see if he is willing to start to pray before getting married? What if he starts to get feelings for her because of the fact that he knows that she is interested in him, will she be responsible for that, or will she not, because it does not involve forming a haram relationship but just letting him know she is interested in him through a third person? To sum up my questions:
My first question is not about whether she can marry him if he does not pray, because her intention is not to marry him if he does not pray. The question is if it is permissble to let this man know that she is interested in him even if he does not pray right now, with the intention to see if there is any chance that he will pray.
The second question is whether she will be responsible for the feelings that he may get for her once he knows that she is interested or likes him, regardless of whether they get married or not.
Please answer as soon as possible. May Allah bless you.

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

First of all, we say that there is no harm on that woman in showing her interest in marrying that young man if he repents to Allah and observes the prayer regularly. Anas, may Allah be pleased with him, narrated, "Abu Talhah proposed to marry Umm Sulaym, may Allah be pleased with them, so she said to him, 'O, I swear by Allah, a person like you should not be refused, but you are a disbeliever, and I am a Muslim. So it is not permissible for me to marry you. If you embrace Islam, then I will consider that my dowry is your becoming Muslim and will not ask you for anything else.' He then embraced Islam, and his becoming Muslim was her dowry.'" [An-Nasaa’i – and he wrote a Chapter on this entitled: "The Chapter of embracing Islam as a dowry for marriage"]

However, she should not accept to marry him simply because he claims that he has repented and that he preserves the prayer; rather, it is more appropriate for her to wait and see until it becomes evident to her that he is truthful in his claim and that he is upright and is religious and has a good moral conduct, because we are guided by the Shariah to choose a person with such qualities.

The Prophet said, "If a man whose religion and moral conduct are pleasing to you proposes to you, let him marry; otherwise, there will be great mischief and corruption on earth." [At-Tirmithi and Ibn Maajah]

It was also reported that a man asked Al-Hasan Al-Basri, "To whom shall I marry my daughter?" He replied, "To the one who fears Allah, because if he loves her, he will honor her, and if he hates her, he will not be unjust to her."

If it is not predestined for them to marry and his heart remained attached to her, then he is not held accountable for his feelings over which he has no control, let alone her being accountable for it, as it is something beyond his control. If we presume that this attachment results in what is not pleasing to Allah, then it is he who will be held accountable and not her, and she is not considered the reason for it such that she would be accountable for it. He is obliged to strive against himself and fear Allah and avoid what displeases Him. 

Allah knows best.