Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure? I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the following all manifolds are 4 dimensional.
I will assume I understand that manifolds where the second Stiefel-Whitney class $w_2 \neq 0$ do not admit a $Spin$ structure (although not clear to me). Let us consider therefore such a 4-manifold. In order to construct something like a $Spin$ structure we need the "square root" of a line bundle $L$ which apparently does not exist (or is not well-defined). 
Question 1: If $w_2=0$ is it well defined? Why do we only care about $L^{1/2}$ now? 
Additionally the spinor bundle $S$ is also not globally defined (and I assume this has to do with $\pm 1$ ambiguities. 
Question 2: Am I right? Is there a nice example where going from one chart to another $S$ (or sections of it) changes signs?
But then, magically, the structure $S \otimes L^{1/2}$ is well defined (because negative signs cancel to each other naively). So now we have this $Spin_c$ structure. 
Question 3:What do sections of this spin bundle look now? In the case of spin manifolds we only had the spin bundle $S=S_+ \oplus S_{-}$. What is the case now? 
I would really like to understand what section of this bundle look like. Finally this yields the final question which is 
Question 4: How is the above related to the fact that $Spin_c(n) = Spin(n) \times U(1)?$ 
 A: There is a ridiculously well written introduction to Seiberg-Witten theory by Moore here. In particular, I think it answers all your questions. Though everything is in the book, let me just summarize a bit. 
Definition:
Basically you can take the following as definitions:
$$Spin (4) = SU(2) \times SU(2) = \left\{ B=\begin{bmatrix} A_+& 0 \\ 0 &A_- \end{bmatrix} : A_\pm \in SU(2)\right\}$$
and 
$$Spin^c(4) = \left\{B^c= \begin{bmatrix} \lambda A_+& 0 \\ 0 &\lambda A_- \end{bmatrix} : A_\pm \in SU(2), \lambda \in U(1)\right\}$$
Note that in particular $Spin^c(4)$ is not the direct product $Spin (4) \times U(1)$ (understanding this is quite crucial in knowing why $L^{1/2}$ might not exists). 
Note also that there is an inclusion $I : Spin(4) \to Spin^c(4)$. 
Homomorphism, actions: 
The $Spin(4)$ has natural projections $\rho_\pm : Spin(4) \to SU(2)$, sending $B$ to $A_\pm$. Also $Spin(4)$ has the following double covering to $SO(4)$: (here we think of $\mathbb R^4$ as the space of $2\times 2$ matrices)
$$\rho : Spin (4) \to SO(4), \ \ \ \rho(B) Q = A_- Q (A_+)^{-1}.$$
Similarly, $Spin^c(4)$ admits two homomorphisms $\rho_{\pm}^c$ to $U(2)$, sending $B^c$ to $\lambda A_\pm$ respectively. Also it has the following double covering 
$$ \rho^c : Spin^c(4) \to SO(4) \times U(1), \ \ \ \rho^c (B^c) = (\rho (B^c), \lambda ^2).$$
here by $\rho(B^c)$ it means we use the same formula as in the Spin group $Spin(4)$, that is,
$$\rho(B^c) Q = (\lambda A_-) Q (\lambda A_+)^{-1}.$$
Note $B^c \mapsto \lambda$ itself is not well-defined, while $B^c \mapsto \lambda^2$ is: indeed $\lambda^2 = \det (\lambda A_\pm)$. In particular, one has the following homomorphism
$$ \det : Spin_c \to U(1)$$
All these homomorphisms of course induces actions on vector space: $\rho_\pm(B), \rho^c _{\pm } (B^c)$ acts on $\mathbb C^2$, $\det (B^c)$ acts on $\mathbb C$ etc.... 
Spin structure, associated bundles: 
A Riemannian $4$-manifold $(M, g)$ can be understood as the following information: there is an open covering $\{U_\alpha\}$ of $M$ and a family of smooth mappings
$$ s_{\alpha\beta} : U_\alpha \cap U_\beta \to SO(4)$$
with some compatibility conditions. A spin$^c$ structure on $M$ is a lifting of $s_{\alpha\beta}$: that is, a family of smooth mapping (with some compatability conditions)
$$\widetilde{s _{\alpha\beta} }:U_\alpha \cap U_\beta \to Spin^c (4)$$
so that $\rho\circ \widetilde{s _{\alpha\beta} } = s_{\alpha\beta}$. Note that this immediately induces to the following: 
$$\rho^c_\pm \circ \widetilde{s _{\alpha\beta} }:U_\alpha \cap U_\beta \to U(2)$$
and 
$$\det \circ \widetilde{s _{\alpha\beta} }:U_\alpha \cap U_\beta \to U(1).$$
The first one induces two complex vector bundles of rank $2$ on $M$ while and the second induces a complex line bundle (called $L$) on $M$. 
The two complex vector bundles are denoted $S_\pm\otimes L^{1/2}$ respectively. The reason is obvious: they are defined by the transition matrix $\lambda A^\pm$ respectively, and $\lambda^2$ defines $L$. So it is important to understand that $S_\pm\otimes L^{1/2}$ are really mere notations, there aren't vector bundles $S_\pm$ and there isn't a line bundle $L^{1/2}$ on $M$. 
Although an oriented four manifold does not admit a Spin structure, it always admit a spin$^c$ structure (the dimension is important here). 
