ऊँची एड़ी चोटी का ज़ोर लगाना

No, 'एड़ी-चोटी का ज़ोर लगाना' is the standard idiom. 'ऊँची' (high) is an optional intensifier used for extra emphasis.

Yes! You can say 'उसने एड़ी-चोटी का ज़ोर लगा दिया' (He gave it his all).

It is neutral-to-formal. You can use it in a job interview or a newspaper, but also with friends when talking about serious goals.

It is masculine. This is why we say 'zor lagāyā' and not 'zor lagāī'.

Usually, it's used positively to show effort. If someone didn't try, you'd just say 'उसने ज़ोर नहीं लगाया'.

In this idiom, it refers to the crown of the head, but because 'choti' also means peak, the double meaning of 'from the ground to the peak' adds to the idiom's power.

'Mehnat karna' is just 'to work hard.' This idiom is 'to work as hard as humanly possible.'

Yes: 'मैं एड़ी-चोटी का ज़ोर लगा दूँगा' (I will give it my all).

Extremely. It's a staple in motivational songs and climax scenes.

It means 'heel'.

Yes, 'पूरी टीम ने एड़ी-चोटी का ज़ोर लगा दिया' (The whole team gave their all).

Yes, especially regarding market competition or meeting difficult targets.

Not at all. It is a living, breathing part of modern Hindi.

'To move heaven and earth' is the closest in terms of intensity.