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1

Is it sufficient to show that we cannot calculate the inverse of the function because it is a many to one function and not a one-one function. It would depend on what you need your function for, but by the standard definition of pre-image resistant, no, it's not sufficient. In the standard definition, it must be infeasible, given $x$, to find any $y$ such ...

2

The inverse cannot be calculated, you're right. However, in can be guessed. In many scenarios, for example cracking hashed password, it's enough for the attacker to know a set of possible inputs which result in certain output (of course if their number is reliable) instead of one certain value. So if the one-to-many function is weak, given the output $y$ ...

7

Hash + digital signature If the hash is not collision resistant, the attacker can produce two messages having the same hash. They'll request a signature on the first and present the signature on the second, a forgery. When second pre-image resistance is violated, this attack becomes much more severe, since now the attacker doesn't need control over both ...

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