The basic rule of handling appear on FIFA laws of the games under LAW 12 – FOULS AND MISCONDUCT
A direct free kick is also awarded to the opposing team if a player commits any
of the following three offences:
3. handles the ball deliberately (except for the goalkeeper within his own penalty area)
So the question is what deliberately means.
The section about Handling the ball explains the list of points the referee should consider before deciding the hand ball was deliberately and awarding a free kick.
Handling the ball involves a deliberate act of a
player making contact with the ball with his hand or arm.
The referee must take the following into consideration:
- the movement of the hand towards the ball (not the ball towards the hand)
- the distance between the opponent and the ball (unexpected ball)
- the position of the hand does not necessarily mean that there is an infringement
- touching the ball with an object held in the hand (clothing, shinguard, etc.) counts as an infringement
- hitting the ball with a thrown object (boot, shinguard, etc.) counts as an infringement.
I also found another reference to talking about a specific case of unintentional hand ball:
If a spectator blows a whistle and the referee considers the whistle
interfered with play (e.g. a player picks up the ball with his hands,
assuming that play has been stopped), the referee must stop the match
and restart the play with a dropped ball from the position of the
ball when play was stopped, unless play was stopped inside the goal
area, in which case the referee drops the ball on the goal area line
parallel to the goal line at the point nearest to where the ball was
located when play was stopped.
A specific answers to your questions:
Is it a foul if a player touches the ball with his hand unintentionally?
Is it a foul is a player trips and falls and touches the ball with his hand during the fall?
If the refree interprets the hand ball as unintentionally so he won't award a free kick.
Also, is a foul awarded if the ball touches the hand of a player, when the player has his hand attached to his chest?
No, because 2 reasons:
1. The ball moves towards the hand and not the other way around.
2. The position of the hand does not necessarily mean that there is an infringement since the hand didn't affect the movement of the ball (it would hit the chest if the hand wasn't there)
Source: FIFA laws of the games