Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.
Should vowed fast be offered in consecutive days, or not?
Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.
Should water be put in the deceased`s mouth and nose while washing him/her?
Putting water in the mouth and nose of the deceased while washing him is impermissible.
Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?
All perfect praise be to Allah,The Lord of The Worlds He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.