The sentences with 'go' don't sound natural to me. 'Go on' means continue, but 'go on with it' is not a phrase which means much. I'd understand what somebody meant if they said this, but this isn't an expression which native speakers would use. 'Get on with it' is a very natural phrase, however. It means : stop talking about it, stop worrying, stop messing around, stop wasting wasting time, and let's just do what we've got to do.

I suppose that these two sentences mean the same. They mean 'Let's (just) continue (it) !'

I agree with Su.Ki. The phrase "get on with it" sounds very natural and is commonly used, however I would change the first sentence to "Let's just carry on (with it)" (meaning let's just continue). "Stop complaining" sounds more natural and works better together with "get on with it" as you have written it, rather than with "carry on" or "go on".