Let M be a smooth manifold and Let G be a Lie group. We have the sheaf $\mathcal{F}_G=(F,\rho)$ of groups defined by $$F(U)=C^{\infty}(U,G)$$ for open sets $U\subset M$ (in particular set $F(\emptyset)=0$ and where $C^{\infty}(U,G)$ is the group of smooth mappings from $U$ to $G$) and $$\rho_{V,U}:F(U)\rightarrow F(V), f\mapsto f|_V$$ for an inclusion $V\subset U$. It is a nonabelian sheaf, but we may define the first and second Čech cohomology with coefficient $\mathcal{F}$ by the following way: Let $C^p(\mathcal{U},\mathcal{F})$ denote the free group of $p$-cochains for an open cover $\mathcal{U}$ of a topological space $X$. For $p=0,1$, define the coboundary $$\delta_p:C^p(\mathcal{U},\mathcal{F})\rightarrow C^{p+1}(\mathcal{U},\mathcal{F})$$ by $(\delta_0g)_{\alpha \beta}=g_{\alpha}^{-1}g_{\beta}$ and $(\delta_1g)_{\alpha \beta \gamma}=g_{\beta \gamma} g_{\gamma \alpha} g_{\alpha \beta}$. Then $\delta^2=1$ and we get the cohomology group $$H^0(\mathcal{U},\mathcal{F})=\text{ker}\delta_0$$ and a simply set $$H^1(\mathcal{U},\mathcal{F})=\text{ker}\delta_1/\thicksim$$ where $g\thicksim h$, for $g,h\in \text{ker}\delta_1$, is defined by $h_{\alpha\beta}=f_{\alpha}^{-1}g_{\alpha \beta}f_{\beta}$ for some $f\in C^o(\mathcal{U},\mathcal{F})$. Taking direct limit, we have the Čech cohomology $H^p(X,\mathcal{F})$ for $p=0,1$, by the same manner.
By the way, the exact sequences of groups $$1\rightarrow \text{SO}(n)\rightarrow \text{O}(n)\xrightarrow{det} \mathbb{Z}_2\rightarrow 1$$ and $$1\rightarrow \mathbb{Z}_2\rightarrow \text{Spin}(n)\rightarrow \text{SO}(n)\rightarrow 1$$ give us the exact sequences of sheaves $$1\rightarrow \mathcal{F}_{\text{SO}(n)}\rightarrow \mathcal{F}_{\text{O}(n)}\rightarrow \mathbb{Z}_2\rightarrow 1$$ and $$1\rightarrow \mathbb{Z}_2\rightarrow \mathcal{F}_{\text{Spin}(n)}\rightarrow \mathcal{F}_{\text{SO}(n)}\rightarrow 1.$$ Therefore it induces the exact sequences of cohomology: $$H^1(M, \mathcal{F}_{\text{SO}(n)})\rightarrow H^1(M,\mathcal{F}_{\text{O}(n)})\xrightarrow{det} H^1(M,\mathbb{Z}_2)$$ and $$H^1(M, \mathcal{F}_{\text{Spin}(n)})\rightarrow H^1(M,\mathcal{F}_{\text{SO}(n)})\xrightarrow{\partial} H^2(M,\mathbb{Z}_2).$$ If $M$ is Riemannian with $\text{dim}M=n$ then we obtain the principal $O(n)$-bundle $O(M)=\bigcup_{m\in M} O(\mathbb{R}^n,T_mM)$ over M, where $O(\mathbb{R}^n,T_mM)=\{u\in GL(\mathbb{R}^n,T_mM):\text{$u$ preserves matric}\}$. Let $g_{\alpha \beta}$ denote the transition function of $O(M)$, and it gives a cohomology class $$g=[g_{\alpha \beta}]\in H^1(M,\mathcal{F}_{O(n)}).$$
Question. In this case, why does $\text{det}g$ coincide with the first Stiefel-Whitney class $w_1(M)$ of $M$?
If $M$ is Riemannian and oriented, then we obtain the principal $SO(n)$-bundle $SO(M)=\bigcup_{m\in M} SO(\mathbb{R}^n,T_mM)$ over M, where $SO(\mathbb{R}^n,T_mM)=\{u\in O(\mathbb{R}^n,T_mM):\text{$u$ preserves orientation}\}$. Let $g_{\alpha \beta}$ denote the transition function of $SO(M)$, and it gives a cohomology class $$g=[g_{\alpha \beta}]\in H^1(M,\mathcal{F}_{SO(n)}).$$
Question. In this case, why does $\partial g$ coincide with the second Stiefel-Whitney class $w_2(M)$ of $M$?
If these questions hold then the exact sequences of cohomology say $w_1(M)=0$ iff $M$ is oriented, and $w_2(M)=0$ iff $M$ is a spin manifold (I want to show them). Any help is welcome.