The implant in your current form is too weak to be truly efficient.
- Stay connected at all times, at most 3-5 minutes of staying disconnected. You need a special, temporary permit that would allow you to access areas not covered by the network (say, for maintenance).
- If you fail to check in, you are forced into coma until authorities locate you. (to prevent accidents, you receive a signal a minute before you enter the coma, so that you may shut down any machines you're operating and lie down on the floor, if, say, local base station fails and forces a whole district into a coma).
- The moment you enter coma, the implant broadcasts the location at much more energy than normally, to reach other base stations / pierce through obstructions. Authorities arrive to pick you up. If it was merely a silly accident (say, a kid wore "silver robot" costume for a party and neutralized the broadcat) you get a citation and are released. If you tried anything illegal, you're taken into custody.
- The government can "deactivate" any citizen like that at any time.
- Tampering attempts with the implant kill the citizen.
- Pregnant mothers are ordered to enter custody until birth when nearing the term (detected by the implant), so no infants are left "unimplanted".
- The implant compares sight of the citizen with database of known locations of implants. A terrorist without an implant merely entering your field of view is reported to the authorities immediately.
That way the terrorists are quite powerless to "recruit". Citizens avoid uncovered areas, and terrorists are quickly captured if they are as much as spotted by anyone. Removing the implant is akin to defusing a bomb, very difficult. Expose it to any light or decrease liquid pressure around, and poof, the patient is dead. If a genuine brain surgery is needed, the implant needs to be deactivated through sending an encrypted authorization code. If, through an accident, your skull got fractured and the implant detects it as a tampering attempt, tough luck.
Now that we have the new&improved implants, let's look at your questions.
Q1: Is there any scientific study that show people will accept false reality over real world?
Quite a few game addiction studies. There are no population-wide studies for the obvious reason: you can't force everyone to play games / enter VR; you can't know if there are people who would be entirely immune. Nevertheless, a large part of the population would possibly tackle it, especially that it's not entirely fake.
If you talk over the phone with someone, you're not talking with a fake person simulated by the piece of plastic by your ear - you use the piece of plastic to communicate with a real person. Same with avatars of live people in the VR - they are living people, only communicating through the machine with you.
Any productive mental work you perform in there is a real work. If you design a building, or develop a mathematical formula or write a program in VR, it's still perfectly usable in the real world. Art, design, music, science, education, entertainment, this all moves to the virtual reality.
Operating machines through virtual reality is perfectly viable too. An excavator that has a camera and remote-control device in place of the operator booth is currently easier to build than one with an actual operator booth. Driven by a worker immersed in VR it will be more efficient than driven by an operator sitting in the seat, by the controls - better comfort, better control, better relaxation, a dozen cameras to provide view from all useful angles, no noise of vibration. The need for actual physical labor drops immensely as most of works can be performed from within VR better. There are still some jobs that require physical presence and fitness - maintenance, security, jobs where versatility of human body still trumps machines. But they are disliked and undesired; good salary but little prestige, something akin to garbagemen.
Savings. Instead of eating whatever awful goop you are assigned from the reserves, you get fed intravenously without feeling it, while eating delicious virtual steaks for free, and never getting fat from them. Since your body hardly moves, less food is needed. Since you occupy a cubic meter of a "sleeper's pod" a kilometer under ground, you don't contribute to overpopulation. You don't need a house or a car, your furniture in your spacious virtual apartment is never worn, and your bills are so minimal you don't even need to work to just get by. You work if you want luxuries: designer's avatars, access to newest movies, custom-designed food, etc. - or any goods "out there". "Public-domain warehouse" of every basic virtual need is free.
You want to stay out of VR? Feel free. Living is expensive out there, there's little to be seen or enjoyed, you can get hurt, tired, poisoned by bad food, get sunburns on your sensitive skin... blah. It's no fun, but whatever rocks your boat. Make sure to pay your premium health insurance rate, because if you catch something out there, you will need to work back your medical bills.
...after a period of time, the security features of the implants will become completely unnecessary.
Q2: Is there any way for the government to put an end to the terrorist without using force? (any case study)
First, the VR is an utopia. Few fanatics would enjoy the savage life of hunted animals. As overpopulation ceases to be a problem (you can fit a population of a billion people in a 1km^3 cube "pod city") they become easier to locate.
Capture (using a robot with tranquilizer dart gun). Put into VR for 30 days, giving a credit of month's worth of a salary of a surface worker. The only restriction is the option to return to the surface. Release after a month... if they want to be released.
Q3: Will the population be able to differentiate between realities? can this be the most humane method of keeping population in check?
Most definitely they will. And I believe it would be quite humane.