Solution:

The killer is Alice.
The number in front of the month's name in the note actually points towards the position of a particular letter in every month.
Second of January = A
Third of July = L
Fourth of April = I
Second of October = C
Fourth of December = E.

Thus, the name is Alice.

Solution:

The murderer is Soumya because 26 Jan is a national holiday in India and the municipality office remains closed.

Solution:

It is not Richard because if it was, Richard 1 and Morgan 3 would both be false, which would make Richard 2 and Morgan 2 both true which is a contradiction.

It is not Travis because if it was, Travis 1 and Travis 3 would both be false, which is not possible.

Therefore, the killer must be Morgan.

Solution:

AMAN

Explanation:
The numbers actually corresponds to the months of the year.

4 - April
5 - May
8 - August
11 - November

Now when you combine the initials of these months, you will get the killer's name i.e. AMAN

Solution:

The man had jumped from an aircraft and when he tried opening his parachute, he was unsuccessful. He hit the ground crushing all his bones and died on the spot. The unopened package found near his body is the parachute.

Solution:

After five minutes, the second friend turned around to see if anyone was there or not. At this moment, he was stabbed on back by his friend.

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.