According to the answer by Korman of this question, every irreducible representation is found inside a tensor product of fundamental representations. So there can't be any missing representations.
"The usual tensor constructions" do not include taking direct summands. They include, for example, taking duals, taking tensor products, taking direct sums, and applying Schur functors. In other words, you can apply all functors that come from functors $\text{Vect} \times ... \times \text{Vect} \to \text{Vect}$ (possibly contravariant in some variables).
Is there a 1-1 relationship between representations of double covers & projective representations of the original group?
Yes, in this case, because the double cover is the universal cover. Both are in turn identified with representations of the Lie algebra.
This is a quite deep and complex topic, which certainly needs a several pages article for a decent intro into it.
Spinors (although informally, they were already in use by the end of the $19^{th}$ century) are attributed to Elie Cartan.
Intuitively (not formally), one can say that they "look" like a kind of generalization of the Euler angles: in the sense that they are used to parameterize and describe generalized rotations (in generalized spaces) in a way reminiscent to the use of the Euler angles in the parameterization of $3d$ rotations.
Cartan's initial idea involved the abstract desription of rotations of $3d$ complex vectors: We consider the complex vector space $\mathbb{C}^3$ "equipped" with the product:
$$\mathbf{x}\cdot\mathbf{y}=x_1y_1+x_2y_2+x_3y_3$$
whith $\mathbf{x}=(x_1,x_2,x_3),\mathbf{y}=(y_1,y_2,y_3)\in\mathbb{C}^3$. Then we consider the set of "isotropic" (i.e.: orthogonal to themselves) vectors characterized by
$$\mathbf{x}\cdot\mathbf{x}=0$$
The set of isotropic vectors of $\mathbb{C}^3$ can be shown to form a $2d$ "hypersurface" inside $\mathbb{C}^3$ and this hypersurface can be parameterized by two complex coordinates $u_0$, $u_1$:
$$\begin{array}{c}
u_0=\sqrt{\frac{x_1-ix_2}{2}} \\
u_1= i\sqrt{\frac{x_1+ix_2}{2}}
\end{array} \ \ \ \
\textrm{or} \ \ \ \
\begin{array}{c}
u_0=-\sqrt{\frac{x_1-ix_2}{2}} \\
u_1=- i\sqrt{\frac{x_1+ix_2}{2}}
\end{array}
$$
Cartan used the term spinor for the complex $2d$ vectors $\mathbf{u}=(u_0,u_1)$. From this, the original isotropic vector $\mathbf{x}=(x_1,x_2,x_3)$ can be easily recovered. He then proceeded to describing the rotations of $\mathbf{x}$ in terms of the rotations of $\mathbf{u}$.
For a more modern ... "skratch" on the ... "surface" of these ideas, the notes:
http://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009/lecture-notes-contents/Ch5.pdf might prove useful.
A classic -and according to my opinion, invaluable- source is the work of Claude Chevalley: "The algebraic theory of Spinors and Clifford algebras", Collected works, v.2, Springer, 1995. The classic point of view (spinors as generalized complex spaces upon which the Pauli matrices and more generally Clifford algebras act) is further analyzed. Some useful references (up to my opinion) can also be found at:
https://hal.archives-ouvertes.fr/hal-00502337/document
http://www.fuw.edu.pl/~amt/amt2.pdf
http://cds.cern.ch/record/340609/files/9712113.pdf
http://hitoshi.berkeley.edu/230A/clifford.pdf
Regarding the intuition thing about spinors. Maybe it would be useful at this point to recall that in Classical physics the description is based upon a "rigid" euclidean $3d$ background i.e. vector spaces and euclidean geometry, upon which calculus is performed and produces the quantitative prediction (which is to be tested against experiment). On the other hand, when quantum mechanics and "quantization" comes into play (in almost all elementary senses of the word quantization), the description of the states of a system is based on vectors living inside Hilbert spaces -often infinite dimensional- upon which algebras of "observables" act. The quantitative predictions are now probabilistic and consist of "spectrums" of eigenvalues of the observables.
When coming to the description of the problem of rotations, the classical physics recipe consists of using the euler angles as parameters i.e. as a kind of $3d$ coordinates leading thus to orthogonal and generalized orthogonal Lie groups. In the quantum picture, the "parameters" are now special vectors of quotient spaces of hilbert spaces, i.e. "spinors", upon which rotations, which are now for example, elements of Lie groups, Lie algebras, Pauli matrices, elements of Clifford algebras etc, act.
Best Answer
Almost nobody does this. There is a discussion in the Wikipedia article on spinor groups but it has too many mistakes and too few references. The only solid math reference I know is in Chapter 5 (freely available from author's webpage here) in
Varadarajan, V. S., Supersymmetry for mathematicians: an introduction., Courant Lecture Notes in Mathematics 11. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 0-8218-3574-2/pbk). vi, 300 p. (2004). ZBL1142.58009.
It is useful if you read this answer in conjunction with my answer here.
To describe $Spin(p,q)$ as a 2-fold cover of $SO^+(p,q)$ ($p>1, q>1$) one has to look at the maximal compact subgroups since they carry all the homotopy information. The group $SO^+(p,q)$ has maximal compact subgroup $SO(p)\times SO(q)$ and $$ \pi_1(SO^+(p,q))\cong \pi_1(SO(p)\times SO(q))\cong H_1\times H_2, $$ where $H_1, H_2$ are cyclic groups (either infinite cyclic or ${\mathbb Z}_2$). In particular, if $h_1, h_2$ are generators of $H_1, H_2$, then there is a homomorphism $$ \phi: H_1\times H_2\to {\mathbb Z}_2, $$ sending both $h_1, h_2$ to the generator of ${\mathbb Z}_2$. It is clear that these homomorphisms are independent of the choices of generators. Let $H< H_1\times H_2$ denote the kernel of $\phi$, it is an index 2 subgroup in $\pi_1(SO(p)\times SO(q)\cong \pi_1(SO^+(p,q))$. Then $Spin(p,q)$ is the 2-fold cover of $SO^+(p,q)$ associated with the subgroup $H$ of the fundamental group. This is your topological description of the spinor group.
In Varadarajan's terminology, here is what's going on. The (unique up to conjugation) maximal compact subgroup of $Spin(p,q)$ is $(Spin(p)\times Spin(q))/G$, where $G\cong {\mathbb Z}_2$ is generated by an element $(a,b)\in Spin(p)\times Spin(q)$, where $a, b$ are the nontrivial central elements of $Spin(p), Spin(q)$ respectively, such that $$ Spin(p)/\langle a\rangle = SO(p), Spin(q)/\langle b\rangle = SO(q). $$ Hence, $(a,b)$ corresponds to the "diagonal" element $(h_1, h_2)$ of $H_1\times H_2$.