The dialogue with Papa
In the first chapter while Haruka is hugging Rei, Mitsuki's phone rang, and she said it was her papa. Whom she was speaking to?

About how Rei faked her death
How she managed to get herself there?
You can see in the second chapter that she was at an height which is twice the height of Yuu.
Also I suppose her holes are fake, so she's just flying in the air near the wall? There no ropes or anything that can show you that she's not pinned at the wall.
I suppose it could be not her body (in the last chapters Rei's right thigh doesn't have the barcode anymore so maybe the first one is a fake, but it could also be just a mangaka's mistake), but still, tricking everybody with a fake corpse (and what kind of detective is Hajime who didn't check the corpse?).

Who killed Eiji?
I didn't understand this too. Mitsuki was in the bathroom and she couldn't go out without Eiji barcode. She also didn't still hear the password "for the one I love" still. The other wolf was dead, according to Hajime.

When Haruka died?
After Mitsuki hear the password she is almost all the time near Yuu until she fainted. And she's wearing her uniform. (next question)

About the corpse hanging down the ceiling in chapter 11
It's Haruka's, right? But Hajime said in chapter 16 that he moved Haruka's body in order to trick the wolf (he was thinking it was Yuu at that point). And being him a detective, shouldn't he have been noticed that it had Mitsuki's uniform on it? That means that only Mitsuki could have been killed her and put her clothes on Haruka's body. But when (question above)?

Who entered in Hajime/Mitsuki's room and attacked them?
How many wolves there are?!? There are people I missed in this manga?

About the wound on Mituski body
Hajime said that she was so intelligent to hide it in chapter 16. How could she realized it? She was just a pawn in Rei's play, and Rei couldn't know she was wounded. It looks a little strange to me.

In reality, no one, although from Mitsuki's perspective it was her father due to being under hypnosis. As I understand it, there are 2 different methods of activating "wolf mode: kill" - one is through a message received cellphone sound, and another is by the words "for the ones I love". So, in this instance Rei used the first method to activate Mitsuki's kill mode. It is later revealed that when Rei was putting barcodes on everyone Mitsuki was already in wolf mode, so one could assume that the hypnosis was still active.

Alright I'm going to argue it's the real body of Rei with fake stakes coming out of it, and that the body is indeed there instead of being an illusion or something. Besides, her being able to move and no one bothering to take her down is crucial to some upcoming events.

How did she get up there? Most likely with help from the bald guy with the scar and Mitsuki prior to starting the game, along with some adhesives or something. The barcode might have been a sticker or something, or could have been covered up later on, not really important.

What kind of detective is Hajime for not checking the corpse? A terrible one. Remember how Hajime failed to check the bald guy with the scar for a pulse before he assumed he was dead? To Hajime's defense her neck was pretty high up there, but they could have worked together to get the body down. This is one of the weakest parts of the story, but it's early enough in the manga for readers to not notice the fallacies.

These two events happened at the same time so I'll combine them. Assuming only 7 people were in the building, Rei must have killed Eiji because Mitsuki was busy killing Haruka. How did Mitsuki escape from the bathroom? She has the wolf barcode that opens every door except the final one.

So, timeline:

Yuu's tied up on ground, Hajime wanders off ->
*Rei kills Eiji and hides behind the door->
Mitsuki leaves bathroom and cuts the power, then proceeds to enter Haruka's room ->
Hajime comes back to Yuu after doing NOTHING (plot device) ->
Yuu and Hajime go to Eiji's room, gets cellphone (Rei is behind door, blocking it with axe) ->
Yuu ties up Hajime, leaves him in room with surveillance ->
Yuu reconnects the power, whereupon Hajime sees Haruka using "prostitute slash" on Mitsuki->
Mitsuki finishes killing Haruka->
Mitsuki puts Haruka's head in the rabbit head->
**Mitsuki passes the rabbit+haruka head to Rei, who stashes it in the room with rabbit heads->
Rei uses the exit to escape->
Mitsuki goes back to bathroom->

~time freeze ends~

Yuu finds Mitsuki in bathroom after coming back from power source.

*(Why Rei? When Yuu was watching the flickering surveillance cameras, he first sees 1.Mitsuki (red herring), 2.Haruka, then 3.Eiji + wolf. 4.Hajime was probably getting a soda. This rules out 1234, 5(bald scar guy) and 6(Yuu himself). Besides, Rei admits on ch.10 page 12 that she was "a wolf, too")
**(It's very very likely that Rei took the rabbit+Haruka head from Mitsuki, stashed it in the room with the rabbit heads and exited the place from the final door, which only Rei had access to. This should cut the time gap considerably, but it's still ridiculous. This of course assumes that Rei at some time had to re-enter the place later unnoticed, perhaps during the 2nd blackout)

Now here's the problem, Hajime never suggests to use Mitsuki's barcode more than once, assuming from the start that every barcode can only open one locked door when it was only proven that Haruka's couldn't open more than one. Secondly, he's the worst inspector of human bodies. There's no way Mitsuki could have completely masked her wound from Haruka's attack, given that there was insufficient medical supplies to even treat Mitsuki's arm wound. What the hell was he doing? From the flashback it seemed like he only looked at her back. If only a certain other detective took his place, one whose name contains an "L"...

Correct. The body(minus head) is Haruka's and the clothes are Mitsuki's. Mitsuki switches clothes with Haruka (and I'll tell you why blow). Chapter 16's flashback of Haruka's corpse makes this confusing.
Best way to explain is again a timeline (along with my first one should clear up the most confusing parts of this manga):

Yuu ties Hajime up and leaves him with unconscious Mitsuki->
Yuu leaves to find a way out, leading him back to the room with the rabbit heads->
Soon after Yuu leaves, Hajime frees himself with a scalpel->
Mitsuki wakes up and Hajime inspects her for a wound from Haruka's attack, sees none->
Hajime is convinced Yuu is the wolf (although later he says he wasn't sure still)->
*Hajime leaves and finds Haruka's body(with her original uniform) and hides it to shock Yuu->
**Hajime hides himself->
Mitsuki sees everything Hajime did on the surveillance cameras->
Mitsuki goes to Haruka's body and puts her clothes onto Haruka (this is to trick Yuu into thinking Hajime is the wolf) ->
Yuu finds Haruka's head in the room with the rabbit heads, heads back to surveillance room->
Yuu arrives at surveillance room and thinks Hajime is the wolf->
***Hajime sees Yuu doesn't have a would from Haruka's attack and realizes Mitsuki is wolf->
Hajime gets stabbed by Mitsuki

*(this part makes no sense, how would Hajime tell if Yuu's shock is caused by Haruka's misplaced body? I bet he didn't plan on Yuu finding Haruka's head, which actually shocked him)
**(why not just wait back at the surveillance room along with Mitsuki just in case she IS the wolf? Afraid Yuu might be "shocked" to see you not tied up as that would ruin your assumption that if Yuu came back shocked it absolutely must be because he discovered Haruka's misplaced body? Then why do you come into the room right after with a freaking 2x4?)
***(holy crap stop jumping to conclusions. What if Yuu really is the wolf and HE's the one hiding the wound? How about you make sure whether there's indeed a wound or not.)

Nobody. Hajime freed himself with a scalpel. He made the story up in order to trick Yuu into admitting that he's the wolf, since this story removes Hajime and Mitsuki leaving only Yuu to be the culprit.
Again, the problem is that Hajime assumes there is only 3 people left in the area, meanwhile Rei has been hiding ever since she killed Eiji. Geez, this guy never stops making stupid assumptions, including the one that gets him killed: "o ya dun worri Yuu im liek supr tight wit de forensics guyz"

Rei's hypnosis doesn't turn Mitsuki into a blind killing machine. Mitsuki was in wolf mode from the beginning pretending to be a rabbit, and you can tell she was genuinely shocked when Yuu first lied about his barcode on his stomach, and realized that she had to kill him after all. So, if she's able to put up an acting job like that there is no doubt that she can be clever. Near the end she becomes more and more destabilized because she thinks Yuu keeps lying to her. My problem with her hiding the wound is not her capacity to do it but the resources required to do so, I mean what can hide a wound as if it never happened? Wolverine's healing factor. That's what.