hostage situation but the gun is a fake

There is no clear answer to your first question. The actual circumstances combined with your cop's training, situational awareness, state of mind, available resources, and tactical judgement will all come into play. You can craft the scene pretty much as you like. What does your story need?

As for the prosecution, it won't matter if the gun was real or not. So long as someone/anyone thought the threat was real, it's legally treated as an actual firearm. There will no doubt be arguments offered in court in an effort to mitigate the seriousness of the "armed" aspect of the offence, but typically such distinctions have little impact, short of the sentencing phase.

Here's New Hampshire's statute for armed robbery.

636:1 Robbery. –
I. A person commits the offense of robbery if, in the course of committing a theft, he:
(a) Uses physical force on the person of another and such person is aware of such force; or
(b) Threatens another with or purposely puts him in fear of immediate use of physical force.
II. An act shall be deemed "in the course of committing a theft'' if it occurs in an attempt to commit theft, in an effort to retain the stolen property immediately after its taking, or in immediate flight after the attempt or commission.
III. Robbery is a class B felony, except that if the defendant:
(a) Was actually armed with a deadly weapon; or
(b) Reasonably appeared to the victim to be armed with a deadly weapon; or
(c) Inflicted or attempted to inflict death or serious injury on the person of another,
the offense is a class A felony, except that if the defendant was actually armed with a deadly weapon, and the deadly weapon was a firearm, he shall be sentenced in accordance with RSA 651:2, II-g.

Notice the emphasized section. It's easier for me to look up the armed robbery statute than to look up kidnapping, which is a complicated statute. The fake weapon loophole was common for a while, and then the law was changed to get rid of it. This was forty or more years ago.

And rarely do I feel a need to correct IronMike, who really knows his stuff. But usually arguing the gun is a fake at sentencing might get you a cup of coffee if you give the prosecutor five bucks when he goes to get coffee. Most likely it won't even get you that good a deal. You're only argument is if the victim thought the gun was fake. Even than it's not likely to help.

Best of luck,

Jim Clark-Dawe

This actually happened in the village where I live. A chap who was on weekend release from prison had beaten up his mother (I'm not sure now, but I think she may have died) and fled. When he got to Hay, a local copper happened to be walking up the street towards him, and the bad guy grabbed the copper and dragged him into the local dentist's waiting room, using a fake gun. There was a siege which went on until the early hours of the next morning, when the bad guy started nodding off, and was jumped by the local lady vet, who had been waiting to have her teeth seen to.
One chap I know had to leave his flat while a police marksman moved in to cover the dentist's windows, and Hay was over run with TV vans - the local chippy did a roaring trade all night!

What do you need for your story? Could range from the cop shooting your bad guy dead to the cop holstering his firearm and telling the bad guy his fake gun isn't convincing with its neon orange paint color.

Whether or not a gun is fake, the crime is the same. He could even commit the crime with no weapon and simply say he had a gun in his pocket. At sentencing, the lack of a working firearm might come into play but, by then, the bad guy is hosed anyway. If you need an out or a lighter sentence, look for other factors to write into the story line.

Jeff

In the USA, "1)" is going to depend on whether the law enforcement organization the cop is part of has an official policy. A police department in a large urban area would likely have an established policy, probably along the line of "de-escalate, disengage, and call in the trained hostage negotiators." For a rural county deputy, it's probably entirely within the deputy's judgement of the situation.

The details of "2)" will vary by jurisdiction, but as others have noted above, it generally hinges on some variant of "credible threat."