What is the ruling on smoking?

All Perfect Praise is due to Allah, The Lord of The Worlds

Smoking is forbidden (Haram) since it is harmful to one`s health as confirmed by trustful physicians, and Islam has forbidden taking anything that causes harm. The Prophet (PBUH) said: "There should be neither harm nor reciprocated harm." [Ibn Majah and Ahmad]. And Allah Knows Best.

What is the ruling on a woman driving a car?

All Perfect Praise is to Allah, The Lord of The Worlds, and may His Peace and Blessinsg be upn our Prophet Muhammad and upon all of his family and compnions.

It is permissible for the woman to drive a car, but it is impermissible for her to travel unaccompanied by a Mahram (Unmarriageable kin). And Allah The Exalted Knows Best.

I used artificial nails on my wedding night, but I was surprised to find out that they require at least a week to remove. However, I performed all my prayers during that period. What is the ruling on those prayers?

 

If these nails prevent water from reaching one's skin then your prayer are invalid and obliged to redo all of them. And Allah Knows Best.

How should a woman make up prayers that she missed two days from delivery?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                        It is prohibited to delay prayer beyond its prescribed fixed time, therefore, the woman in question is obliged to make up missed prayers after delivery and confinement i.e. once she becomes ritually pure. And Allah Knows Best.