Why can characteristic polynomial be calculated using $\det(A - \lambda I )$ or $\det(\lambda I - A)$ Wouldn't the two formulations give you completely different eigenvalues?
Seems dangerous and callous to me.
 A: Remember that for any $n\times n$ matrix $X$ and any $r\in K$, with $K=\mathbb{R}$, $\mathbb{C}$,or any commutative ring, as @MarcvanLeeuwen points out. Then we have that:
$$
\det (rX) = r^n \det(X).
$$
In particular, if $r=-1$, you get 
$$
\det(-X)=(-1)^n\det(X).
$$
So the roots of both of them are the same and their respective characteristic polynomials will only differ by a sign.
A: For an $n\times n$ matrix $A$, those two determinants will be the same, except for a change of sign if $n$ is odd.  Since you are finding the eigenvalues by setting the determinant equal to zero, the change of sign does not matter.
A: No the two determinants differ by a sign $(-1)^n$ where $n$ is the size of the matrix, and (for odd $n$) only one of them is the characteristic polynomial. Which one depends on the book you are using, but I think there is preference to defining it to be always monic, in other words taking $\det(\lambda I-A)$ as the definition of the characteristic polynomial.
This does not prevent either one or the other to be used to have a polynomial whose roots are precisely the eigenvalues;  alternatively one could choose to use neither and find eigenvalues as roots of the minimal polynomial of$~A$. In fact there is no absolute link between the notions of eigenvalues and the characteristic polynomial: both are defined independently. So the chosen definition for characteristic polynomial does not affect the set of eigenvalues.
If all one does with the characteristic polynomial is to find the eigenvalues as its roots, then either definition will work equally well. 
However, there are more advanced applications, for which the characteristic polynomial always being monic is really more convenient (as always, one can make do with inconvenient conventions, like saying "when the minimal polynomial and characteristic polynomial agree up to sign"; only such work-arounds distract attention from what is essential).
