1) Will i have an impact on "car valuation because of over due NCT'?
Yes, I imagine that the assessor would take that into account.
2) What will be the realistic time for settlement in Ireland?
I'm only familiar with one example. A crashed into B. A's assessor was on the scene 2 days later, A's insurer made an offer 2 days after that, and a cheque was in the post maybe a week after that. That's the best case scenario - "sometime next month" is probably the best you can hope for, assuming the car isn't found alive (you're probably better off if it isn't).
3) will it impact my new insurance?
Afaik for most policies, fire and theft claims don't count against your no claims bonus. You'd have to check your own policy to be sure.
4) My insurance is due in July. Should I revenue it till the settlement happens?
Check with them, but I don't think it's necessary to renew. The car was insured when the loss happened, so the insurer is on the hook regardless of what happens to the policy in the future.