the first chern class of complex vector bundles

Question

Let $\xi^\mathbb{C}$ be an $n$-dimensional complex vector bundle over a manifold $M$, $n\geq 2$.

Question 1: Are there any practical methods to detect whether the first Chern class (with integer coefficients, NOT with rational coefficients) $$c_1(\xi^\mathbb{C})$$ is zero or not?

I have found two ideas:

(I). The first idea is given in the mathoverflow question as in the following picture.

Question 2: Where to find the references giving the formula $$ c_1(\wedge ^n \xi^\mathbb{C})=c_1(\xi^\mathbb{C})? $$

(II). The second idea is given in Vanishing of the first Chern class of a complex vector bundle.

Question 3: Any references giving whether the following is true or not? "The first Chern class $c_1(\xi^\mathbb{C})=0$ if and only if the structure group of $\xi^\mathbb{C}$ can be reduced to $SU(n)$."

But I think the answer is about rational coefficient Chern class, not integer coefficient as we want.