Applying for a new government job

You are already working in the Govt sector and have completed 3 years of service. As per the rules and regulation your department should forward your application for the post in other Govt department. I do not see any reason why they should not do that. You will have to meet the top boss or the secretary and request for this through proper channel forward of your application.
If you hide this fact then you would be in problem later as there would be some problem in your PF and NPS accounts. Moreover if you directly apply and hide your present job then you would be treated as a raw person without experience and would lose merit for selection in the new place. Another problem is that if you hide this fact and get selected in the new position then you would have to resign immediately from the present post and for that you might have to forego your one month salary or whatever is there in the contract in your job. As per the rules one is supposed to give a one month notice for resigning and in your case if you get selected and asked to join immediately then you do not get time gap to give notice.
Please try to get the forward of your application and NOC from the present department without which you would be in problem.
I would suggest you to make a case file of all the correspondences between you and the present department and fight your case strongly by writing reminders and if you are keeping your records properly it would help you in many places in future.

1. No. You should not do that.
2. Giving wrong information or hiding facts will be a crime and your application will be considered by the government office. They may remove you from services the moment they come to know that you have hidden the facts.
3. No office will entertain the application of a government service which was not sent through the proper channel or no NOC is produced.
4. There are chances they can inform the present office where you are working and they may go for disciplinary action against you as per the procedure.
Keeping the above facts in view you should not send your application without informing the present employer.
What are the conditions of your present employment? There is no way they can deny NOC to you if there are no issues with you. So you better approach the boss and explain to him the requirement and then if he says he can't give it, you can represent the matter to the grievances committee and try to get the NOC.
WIthout showing the experience if you apply, you will lose your seniority and also some problems will come with your PF account.

If you join the new assignment without intimating your present status, the following would be the ultimate outcome-
1) You will be loosing the benefits of the working experience which can boost your career prospects by way of your further promotions/ suitable pay hikes for such experience acquired in the past.
2) There will be breach of accumulation of money in your NPS account which in due course will impact you heavily in terms of accumulation of your deposits.
3) There might be the chances of leaking of your past history and in that case, your present employer can terminate you for the suppression of facts.
4) Punishment could be severe to the extent that you will be jailed.
The best recourse for you would be as follows -
1) You should intimate your present employer that you want to move this organisation due to your selection of the executive post in the government. Your application should also to be routed through the proper channel and for that you must secure NOC.
2) You maintain transparency at all costs. This would be applicable for both the organisations- the present one and the next one where you want to join.
3) You should intimate the next employer regarding your probable date of your joining.If there is any chances of delay in your joining, the same is to be intimidated with the detailed reasons for the delay which might be beyond your control.
Maintaining the continuity of service has several advantages such as your experience for rendering your different jobs would be clubbed together. Even your gratuity amount in your final settlement would be more than your breached service conditions. Your NPS amount and PF accumulation will be substantially higher than any discontinuation occurring in between two services.