سوال ہی پیدا نہیں ہوتا

It can be, depending on your tone. With friends, it's fine. With a boss, it's very bold. Use it when you need to set a firm boundary.

Usually no. It's almost always used for negative refusal or impossibility. However, you can say 'My love for you is so strong, the question of leaving doesn't arise.'

'Namumkin' is just a fact (impossible). 'Sawaal hi...' is an emotional and emphatic reaction to that fact.

Yes, Hindi and Urdu speakers both use this phrase identically, though the script differs.

Yes: 'Sawaal hi paida nahi hota tha' (The question didn't even arise).

In this context, it means 'to arise' or 'to come into existence'.

Yes, that's a common short version, but the full version is more powerful.

It's an emphatic particle, like saying 'The question *itself*'.

Yes, in editorials or opinion pieces to dismiss an idea.

Yes, the phrase is gender-neutral because the verb 'hota' agrees with the masculine 'Sawaal', not the speaker.