Is it true that whatever is dry is pure even if there was impurity on it?
The impure material object does not soil (make impure) anything else unless the impurity is wet and transferable. But, if the impurity is dry or non-transferable, then there is no problem. Fuqaha has stated that: There is no disagreement between Muslim scholars that when a dry impurity touches something dry it does not make the latter impure.
Is touching a non-Mahram woman invalidates person's ablution?
Touching non-Mahram woman is prohibited, and according to the Shafiee's, it invalidates the ablution. And Allah Knows Best.
What is the ruling on a Muslim who slaughters an animal while being in a state of Janabah (ritual impurity)?
All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upo all of his family and companions.
Slaughtering is permissible for the ritually impure man and the menstruating woman because ritual purity isn`t a condition for the validity of the slaughtering. Moreover, it is permissible for the uncircumcised person to do the slaughtering as well. And Allah The Almighy Knows Best.
Is it permissible to pray with a heater (stove) in front of me?
There is no harm in praying with a heater in front of you, as long as you do not intend to venerate the fire or imitate those who worship it. And Allah Knows Best.