Wa 'Alaykum-us-Salaam wa Rahmatullaahi wa Barakaatuh.
Firstly, it is wrong for you to have contact with the girl before Nikaah. Islam does not permit any form of contact between a non-mahram male and female prior to Nikaah. You should not be talking to her or discussing anything with her directly, if at all you do plan to marry.
* Abu Hurayrah (radhiallaahu anhu) narrates that the Messenger (sallallaahu 'alyhi wa sallam) said: A woman is married for one of four qualities. Either for her wealth, or for her pedigree (family rank and standing), or for her beauty, or for her Deen (piety). So marry a pious lady and achieve success. (Mishkaat ~ 267, Agreed Upon)
* Abdullaah ibn Umar (radhiallaahu anhu) narrates that The Messenger (sallallaahu 'alyhi wa sallam) said, " Everything of the world is worthy of (temporary) benefit and the most benefical (enjoyable) medium of this world is a pious lady." (Mishkaat, Agreed Upon)
* The Messenger (sallallaahu 'alyhi wa sallam) commented: "Do not marry women (merely) on account of their beauty...for it is possible that this very beauty may become the cause of her destruction. Niether marry women (merely) because of their wealth...for it is possible this may be a cause for her rebellion and mischief. Rather marry women because of their religiousness (piety). (Remember!) A slave, dark in complexion, graced with piety and noble character is infinitely superior to a beautiful, highclass woman of poor character." (Ibnu Maajah, quoted in Uswa Rasool Akram)
* Abu Umaamah (radhiallaahu anhu) narrates that The Prophet (sallallaahu 'alyhi wa sallam) said, "No man can derive more good for himself from anything after the fear of Allaah, than a pious wife; who obeys him when he instructs her, who pleases him when he looks at her, who fulfills his oath when he swears an oath (that she should do something within the bounds of Shari'ah), and who if he is away from her, safeguards her chastity and his wealth. (Ibnu Maajah)
1) From the aforementioned Ahaadeeth, we learn very important lessons:
-> When selecting a spouse, the most important thing that should be taken into consideration is piety and religousness.
-> Wealth, beauty and pedigree are all fleeting qualities.
-> A successful marriage and a successful life in both worlds is achieved by marrying a pious woman.
-> One of the most important traits of a wife is that she is obedient to her husband in all that he instructs her to do, providing it is within the limits of Shari'ah.
If she is unwilling to listen to you before marriage, there is little hope that she will do so after marriage.
MashaAllaah, you are a Haafizh of the Qur'aan and an Imaam, obviously conscious of religion and your responsibilities towards Allaah. It is important for you to find a suitable girl.
A Persian couplet:
"Should the builder lay the foundation incorrectly, will arise the wall, even it be as high as to the heavens, crookedly!"
2) It is wrong for you to have unnecessary contact with her. However, on the basis that she is a marriage prospect, and you would like to ascertain her religious outlook and at the same time make your outlook clear to her, your asking her to cover her face is definitely not wrong.
We would nevertheless advise you to perform Istikhaarah Salaah and earnestly ask Allaah to decide what is best for you in terms of Deen, life in this world, and in the Hereafter. The method of performing Istikhaarah can be learnt from the following link:http://www.cislamonline.com/library/index.php?view=marriage/istikhara
And Allaah knows best.