In the book I'm reading they say they want to deduce $(p \rightarrow q) \rightarrow (p \rightarrow r)$ from $p \rightarrow (q \rightarrow r)$.
Now, as far as I understood, $p \rightarrow (q \rightarrow r)$ is an assumption, and $(p \rightarrow q) \rightarrow (p \rightarrow r)$ is the conclusion.
This conclusion can apparently be taken apart to get auxiliary assumptions. This is done by taking the antecedent of $(p \rightarrow q) \rightarrow (p \rightarrow r)$ as an auxiliary assumption. Then they take another auxiliary assumption (but from where I do not know because it doesn't say) to have p as an auxiliary assumption.
My question is this: if an antecedent has been taken as an auxiliary assumption, can it be used again to take another auxiliary assumption? So for example, from $(p \rightarrow q) \rightarrow (p \rightarrow r)$ we take $(p \rightarrow q)$ as an auxiliary assumption, and is now in the assumption collection. But can we now take p from $(p \rightarrow q)$, or can we only take it from $(p \rightarrow r)$.
Second question is, can we only take the antecedent as an auxiliary assumption? So q can never be an auxiliary assumption?
In standard natural deductions systems, you can make any new assumption you want to at any time. (Often this is called the 'Rule of Assumptions' or the like.)
Of course if you are trying to show that some given assumptions $A, B, \ldots, C$ entail a given conclusion $E$, then any additional assumptions you make along the way -- auxiliary assumptions made "for the sake of argument" -- will need to be discharged.
For example if the conclusion $E$ is of the form $p \to q$, it will often help temporarily to assume along the way $p$, deduce $q$, and then at the end discharge that additional assumption to infer as wanted $p \to q$ from just the original premisses. For another example, if the conclusion $E$ is of the form $\neg p$, it may well help temporarily to assume along the way $p$, deduce a contradiction, and then at the end discharge the assumption we've reduced to absurdity, and infer as wanted $\neg p$ from just the original premisses.
But yes, it is liberty hall in natural deduction systems -- assume whatever you want, whenever you want, keeping track as you go of what assumptions are in play. Though of course what you deduce will then depend on not just the original premisses (if any) but on any additional assumptions in play, until such time as they get discharged using a rule like Conditional Proof or Reductio ad Absurdum.