Is it permissible to perform istinja' directly under the water spray in the bathroom, and with which hand should one perform istinja' and pour water, and is it obligatory to perform istinja' three times?

Istinja' is the removal of what exits from the two private parts with water or a stone, etc. The left hand is used for istinja', not the right, unless there is an excuse. Pouring water when needed is done with the right hand. Thorough cleansing is obligatory in istinja'. If done with water, it is sunnah to wash three times. If done with a stone or the like, no less than three wipes are sufficient.