Solution:

Molly

Explanation:
Let us suppose that Patrick was not in the second bedroom. This proposes that Jerry, Molly, Max and Mike are liars. It infers that Patrick and Catherine speak truth. But that is not possible as they do not coincide on Mike's location. Therefore, we are wrong with our assumption and Patrick was in the second bedroom.
So far, we have established the fact that Patrick was in the second bedroom. Additionally, Patrick and Catherine are liars

Now, let us suppose that Jerry speaks truth. This proposes that Max was in the bathroom and Jerry in the second bedroom. This infers that Molly and Max are liars as they told that Jerry was not in the second bedroom. Also it implies that Mike lied, as he said Max was in kitchen. Therefore we have three new liars now, which totals the number of liars to 5, which contradicts the given situation in the question.
Hence, now we have established the fact that Jerry is a liar.

Let us now suppose that Molly speaks truth. This proposes that Jerry was in the living room and Max was in the Kitchen. This infers that Max and Mike are liars. But again, this is contradicting the situation as we have already established the fact that there are again 5 liars (counting above 3 established liars)
Therefore, now we have established that Molly is a liar. Max and Mike speak truth. Catherine was in the bathroom. Jerry was in the first bedroom. Max was in the kitchen. Mike was in second bedroom with Patrick.

The killer has to be Molly as no one else was present in the living room.

Solution:

It is simple. The sheriff never told him where the crime scene was. But still he reached at the right place. This points towards the fact that he knew it already and that he is the killer himself.

Solution:

The wife booked a two way ticket for herself and booked a one way ticket for her husband.

Solution:

The dead man used an ice knife to commit suicide. The ice held the knife pointing upwards as he dropped on the chair with his back facing the knife. Later, the ice melted leaving moisture around the body.

Solution:

It is quite possible. The poison was not on the drink but in the middle of ice cubes. The first one drank immediately and thus the poison could not reach his body while when the other drank after half an hour, the ice cubes had melted. Thus he died.

Solution:

Bob was in China in 2012 & he killed Anderson.

Anderson died of suffocation=>This information should lead you to statement #3 next. From here you should gather that Bob killed Anderson.
Thus leading to statement #4, Bob was in China in 2012.

Solution:

1)The first bit is easy, as the first letter of each plate spells WHODUNNIT IX(A challenge to our Detective.)
2)The second bit is a little trickier, but I gave you the solution.If you read the last three letters in each plate from the bottom up and right to left you get ADVANCE EACH LETTER BY THE NUMBER SHOWN, so advance W by 5 to get B, H by 13 to get U and so on until you spell BUTLER DID IT.

Solution:

Dabid is the killer

drawing room(D) , art room(a) , bed room (b) , ice-cream room (i) , desk room (d)

Solution:

She was hoping that the guy would appear at the funeral again.

Solution:

Newspaper delivery man was the murderer because of absence of Wednesday and Thursday missing newspaper.
This indicates he already know that there is no one to read it.