1. Suppose someone asks me 'Are you in queue?' when in fact I'm not because I'm just standing in the line. Is it natural and polite to say 'Not really'? If not, what is the polite reply?

2. When is it appropriate to reply with 'not really'? Could you please give examples?

3. In the first place, are saying 'no' and 'not really' the same in meaning or intent? If not, how are they different?

Thank you.
To me, you are either in line (queue) or you are not. " not really " is a mixed message. But I will take it as " not " personally.

Q- Is this dish very hot ?
A - Not really! (Suggesting it is hot to some degrees, but not very hot to the answerer). That's how it sounds to me.
Thank you, dimusumexpress, for sharing your thoughts on this.

I was wondering if a native speaker could also share his/her thoughts on my questions. Thanks.
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I think Dim Sum is right on this one. You are in the queue or not, its a yes or no. I also think his example is a good one.

Not really is sometimes used as a no but it doesn't really mean no. It means somewhere between yes and no but near to the no. If asked if I'm hungry I sometimes say not really, as it means not I'm not hungry but I would be able to eat something.
OK, thank you, Dave. That was really helpful.

Just one more question, in my original example, would you agree 'not really' is not an appropriate answer because I'm not in cue at all?

Can I say instead, 'No, I'm not in cue. Go for it'?
Queue not cue. A cue is what we play snooker with. I would simple say no and move out of their way. If you say anything more you might be in danger of the queue continuing but you yourself not realizing it.
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OK, thank you for your advice and for pointing out the difference in spelling. I really appreciate them.