في أقرب وقت

No, it's perfectly fine, though 'soon' (qariban) is more common if there's no real rush.

No, it is strictly for future actions or requests.

'Aqrab' means nearest (time-wise), 'Asra' means fastest (speed-wise). They are often interchangeable.

You can use the same phrase, or say 'fi asra' waqt' or just 'Gawam'.

Not at all. It is considered polite and standard. To be even softer, add 'law samaht' (if you please).

Because it's a superlative-noun construction where the noun remains indefinite to show 'any' nearest time.

Yes, it's great for saying 'I can start ASAP'.

The words 'qarib' and 'waqt' are, but this specific three-word idiom is a later development of the language.

There isn't a direct idiom, but you might say 'fi waqt lahiq' (at a later time).

No, for distance you would say 'aqrab makan' (the nearest place).