During the hearing (in OotP) Fudge mentions that "...there are no dementors that are not in our control...", so yes, maybe they have a count and identification. However, we need to look into some discussions that happened in PoA.
- Dumbeldore mentions "... it is not in the nature of Dementors to forgive ..."
- Fudge mentions that Dementors tried to kiss Harry (w/o authorizations) and Sirius. So he had to send the dementors back to Azkaban.
So, if we look into these points:
Does Ministry have a record of Dementors?
Yes (mostly), but after Voldemort come back to life, the ministry loses control over them. Later they were disbanded.
How many are there in the country? Who's placed where? How do they identify one from another?
Again, yes they had. From Harry Potter wikia [1, 2]:
Post 1998
After the Second Wizarding War, the Ministry of Magic became headed by former-Auror and Order of the Phoenix member Kingsley Shacklebolt. Due to their actions during the war (allowing Death Eaters to escape back to Voldemort and joining forces with him themselves), Shacklebolt disbanded the Dementors as guards of the wizard prison. It is unknown where they went or what they did without their only known application. After the Dementors were removed from the prison, Aurors were placed in the prison as guards.
Also, "They cannot be destroyed, though their numbers can be limited if the conditions in which they multiply are reduced, implying that they do die off eventually."
In a case like above, if any dementor (if they are identifiable from one another) breaks the law and kisses an innocent, what punishment will he be likely to receive?
No, I don't think they 'can' be punished like that. Like in PoA when they tried to administer the kiss they were just sent back to Azkaban.
Another thing, in OotP the Demenotrs that attacked Harry and Dudley were sent by Umbridge. So, there was no point in punishment.