Did or did not the murders happen in reality?

The Murders for which Malcolm was arrested did happen and Malcolm did indeed commit them. The plot of the identity is sort of a combination of Fight club along with the famous Agatha Christie novel namely "And Then there were none".

Malcolm committed the murders but he has a split personality disorder and there are 10 personalities to him, Dr. Mallick tries to convene all his personalities in a single place that happens to be the motel as part of the experiment to isolate the killer personality who is actually causing Malcolm to commit the murders he did in the past.

Every event which happens in that motel is happening in Malcolm's head and it includes only his 10 personalities, the other set of events are the actual events with the real Malcolm who is being presented to the jury the day before his execution by the Doctor to plead to them on the basis that its not Malcolm who did those crimes it was an aspect of his split personality with in him.

Dr. Mallick's idea is to eradicate the killer personality because he was the one who committed the murders but the other personalities were completely innocent it includes even the convict Rhodes with the help of John Cusack's personality, who appears to be the most sane personality among the others because of him being a cop prior.

(Unfortunately they zone in on the wrong personality and the killer personality happens to be the little kid who is never suspected throughout the movie. Malcolm is now rid of all his good elements and is completely taken over by the evil personality, which is why he proceeds to kill the doctor.)

No, the murders did not happen in reality (or at least not in a physical way). The whole of the movie's story actually pans out solely inside the head of Malcolm Rivers, a convicted murderer under treatment by Dr. Malick.

Those 10 people are all representations of the different persona Malcolm houses inside his personality (the title of the movie is already a hint to that, as well as other little "mysteries", like everbody having the same birthday date, that of Malcolm Rivers). This whole story of those 10 different persons ending up at the same place and murdered until only one is left is actually a mental representation of Dr. Malick's treatment of Malcolm, whose purpose it is to eliminate all those multiple persona in Malcolm (hopefully also the one that let him commit the murders for which he ended up in custody) until only one is left, Malcolm's true identity. This is actually explained at the end of the movie by Dr. Malick, before which the audience only got some little hints and some snippets of that real scene thrown into the action of the imaginary motel story.

And at the very end it's also revealed that the true evil identity of Malcolm was actually the little boy, who has also survived the motel incident and comes back to murder Paris. And this also coincides with Malcolm "freaking out" in the real world and murdering Dr. Malick and his associates while riding with them in a car.

I would add to the other answers that it appears the scenario in Malcolm's head is based on elements of his real life; he was abandoned in a motel as a boy, and the characters that were manifesting in his mind bore a kind of resemblance to the victims whose photos we briefly saw when his file was shown (not necessarily an exact physical resemblance, but perhaps number, age and gender).

He committed four murders in actual physical world for which he was being tried. The murders at motel however were happening only in his mind. Ultimately the real killer in him the child dominated . In the head the child in him killed the lady and in actual world he killed his doctor in the black out as if the child in him was killing the lady. Poor doctor. Trying to do good, he himself got killed.

I got the movie in a different way. The murders didnt happened in real world.They were a kind of a story made by Malcolm which included the characters whom he lived with (the ten people which he would become due to his illness).

There were only 6 people who were actually killed by Timmy, (The actress , the convict, the couple-husband and wife, his mom and the lady). The rest all died due to the other characters (the ex cop and convict cop killed each other, dad died due to the manager, convivt cop killed the manager). These were the 10 people. Timmy didnt die. Now while Timmy was killing these 6 people in the story going on in the head of Malcolm, Malcolm actually killed 6 people in the real world (5 + 1 the doctor).

The way I read it, Malcolm Rivers was a murderer, but a mentally deranged one because of the abuse and neglect he suffered as a child. All that detail is provided at the beginning of the film.

The murders didn't happened in real world.They were a kind of a story made by Malcolm which included the characters whom he lived with (the ten people which he would become due to his illness).

There were only 7 people who were actually killed by Timmy, (The actress , the convict, the couple-husband and wife, his mom,dad and the lady). The rest all died due to the other characters (the ex cop and convict cop killed each other, convict cop killed the manager). These were the 10 people. Timmy didn't die. Now while Timmy was killing these 6 people in the story going on in the head of Malcolm, Malcolm actually killed 6 people in the real world (6 + 1 the doctor).