The face is the front of an animal's head that features three of the head's sense organs, the eyes, nose, and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities affects the psyche adversely.
No. It is not permissible to display or sell a picture of a female exposing her hair and with make up on her face.
Such picture will promote wrong attitude against women which is used by the sinners who look at the female a source of sexual amusement. Woman is much higher and more noble than to be used as a piece of artwork.
Drawing an incomplete human figure is allowed.
Adding a head with out eyes, nose and mouth is also allowed.
We have narrations from Ahlul Bayt (AS) stating that it is recommended after Du's to wipe your hands on your face. Your hands seeking from Allah in Du'a will never be empty from the gifts and bounties of Allah. Wiping on your face will make your face blessed as well. This is not during Salah.
We don't have any authentic evidence on wiping hand after reciting Sura Al-Fatiha. Some people do it just because it is been done by Sunni brothers who read Sura Al-Fatiha like a Du'a. It has no authentic evidence neither in our books nor in Sunni books.
It is not permissible for the female to do that, and not permissible for a man to look at her without proper Hijab.
Looking at the face of a female which causes lust or attraction is definitely not permissible in Islam. Muslim man must avoid any looking which causes lust or attraction. Lowering gaze is obligatory on Muslims as we read in Quran (Sura 24, Verses 30 and 31.
Yes it is allowed to paint as far as it is not statuary or sculptural.
No, it is not mandatory, but only required.
And Allah knows best.
It is Makrouh (disliked) to pray in front of a mirror. The reward I of the prayer will be less but it is not Haraam to do that. Your prayer is valid even with being in front of a mirror. Makrouh is not forbidden.
Neither washing nose is obligatory in Wudhu nor washing mouth. Both of them are just recommended and not obligatory. Washing head or feet in Wudhu is wrong because you should wipe over them and not wash them.
If you are sure that your Wudhu was invalid, you need to perform it properly and re do the prayers which were done with wrong Wudhu.
The direction of Karbala from India is very near to the direction of Qibla from India. You can refer to local Ulama in your area for more details.
If it is not considered as statues or part of statues then it is permissible.
Drawing a sketch of the face only of your wife is permissible but it is should not be displayed in front of non Mahram men who know your wife.