A woman didn't fufill fasting of the month of Ramadan two years ago, due to pregnancy and breastfeeding, at the time being she is making up the missed Ramadan. What is the ruling of Sharia? And what is due on her? 

Is it permissible for a woman in her waiting period (Iddah) after her husband's death to leave her house and mix with non-Mahrams?

It is not permissible for a woman in her waiting period (Iddah) after the death of her husband to leave her house except for a necessity, such as for medical treatment, nor is she allowed to mix with non-Mahrams. A woman is not permitted to mix with anyone other than her husband or Mahrams, whether she is in her waiting period or not. However, there is no harm in accepting condolences from those who visit and responding to their inquiries about her well-being. And Allah Knows Best.

Is the prayer of a woman who is led by an Imam via T.V. permissible?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.                                                                                                                                                                                                                          Amongst the conditions for the validity of prayer is that both the Imam and those led by him must be in the same place. Thus, the prayer of the person who is being led by a televised Imam is invalid unless he/she was with him in the same mosque. And Allah Knows Best.

How should I deal with the whispers of the devil with regard to ablution and purity?

Whispers in ablution are from the devil, and so you should take refuge in Allah from the devil`s evil suggestions that aim to distort the religion. Thus, once an organ is completely pure, don`t question its purity since having whispers doesn`t mean that the purity is invalid as it has already been established.