The spin-statistic theorem holds in all dimensions of spacetime greater than two. The proper definitions of "half-integer" or "integer" spin in general dimension is simply how the rotation operator of a full rotation, $R(2\pi)$ is represented - "integer spin" or "bosonic" representations will have it as the identity, while "half-integer" or "fermionic representations will have it as minus the identity. More formally, fermionic representations are those that are representations of the double cover of the Lorentz group that do not descend to representations of the actual Lorentz group. If you examine the proof of the spin-statistic theorem you will see that it is indeed the behaviour of the $2\pi$ rotation that is crucial, not that spin is "half-integer".
Talking about "spin up/down" in other than 3+1 dimensions is indeed difficult in general representations. We are used to talk about spin in 3+1 because the Lorentz algebra $\mathfrak{so}(1,3)$ has an accidental equivalence for its finite-dimensional representations to $\mathfrak{su}(2)\times\mathfrak{su}(2)$, i.e. two copies of the rotation algebra. The algebra of actual physical rotations embeds diagonally into these, and we label a finite-dimensional representation of the Lorentz algebra thus by two spins $(s_1,s_2)$ where the total physical spin is $s_1+s_2$. The Dirac spinor in 4D is $(0,1/2)\oplus(1/2,0)$, the sum of the left- and right-handed Weyl spinors.
However, the Dirac representation does have a notion of spin up/down - you can group the Clifford algebra of the gamma matrices into pairs of raising/lowering operators (with the last one remaining unpaired in odd dimensions and being akin to a parity operator), and the Dirac representation is the unique (or, in odd dimensions, one of the two unique) $2^{\lfloor d/2\rfloor}$-dimensional irreducible representation of this algebra, in which the basis states are labeled by "spins" $s_1,\dots,s_{\lfloor d/2\rfloor}\in\{\pm 1/2\}$ and the $i$-th apir of raising/lowering operators raises/lowers $s_i$. For more on constructing the Dirac representation in arbitrary dimensions see this question and this question.
To answer the confusion between the three sources you list:
Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor representations for $p - q \pmod 8 = 1$.
Pinor representations induce spinor representations (that will be reducible in even dimension) and so we get Majorana spinor representations for $p - q \pmod 8 = 0,1,6,7$.
Although $\mathcal{Cl}(p,q)$ is not isomorphic to $\mathcal{Cl}(q,p)$, their even subalgebras are isomorphic and so can be embedded in either signature. This means that Majorana pinor representations in $\mathcal{Cl}(q,p)$ also induce spinor representations in the even subalgebra of $\mathcal{Cl}(p,q)$ and so we also get an induced Majorana spinor representation for $p - q \pmod 8 = 2$
(from $q - p \pmod 8 = 6$; this is often called the pseudo-Majorana representation).
Fecko has his signature convention swapped compared to Figueroa O'Farrill, and so swapping back we see that his $0,2 \pmod 8$ gives us $0,6 \pmod 8$. One can also see from his table (22.1.8) that on the page you reference he was listing signatures with Clifford algebra isomorphisms to a single copy of the real matrix algebra, but his table also gives us $p - q \pmod 8 = 1$, converting signature convention to $p - q \pmod 8 = 7$ which is the isomorphism to two copies of the real matrix algebra and so also yields Majorana pinor representations. He doesn't talk about Majorana (or pseudo-Majorana) spinor representations here and so doesn't list $p - q \pmod 8 = 1,2$.
As for Polchinski, he includes pseudo-Majorana representations (or is signature convention agnostic) and so lists all of $p - q \pmod 8 = 0,1,2,6,7$.
To answer the question of in which dimensions Majorana spinors (including pseudo-Majorana) exist:
For a signature $(p,q)$ they exist whenever any of $\mathcal{Cl}(p,q)$, $\mathcal{Cl}(q,p)$ or the even subalgebra of $\mathcal{Cl}(p,q)$ are isomorphic to either one or a direct sum of two copies of the real matrix algebra. This means $p - q \pmod 8 = 0,1,2,6,7$.
If one discounts pseudo-Majorana spinors, then one removes $\mathcal{Cl}(q,p)$ from the previous statement and this means $p - q \pmod 8 = 0,1,6,7$.
Of course, this does not talk about the naturally quaternionic symplectic and pseudo-symplectic Majorana representations.
One can take the algebra isomorphisms of low-dimensional Clifford algebras ($\mathcal{Cl}(1,0) \cong \mathbb{C}$, $\mathcal{Cl}(0,1) \cong \mathbb{R} \oplus \mathbb{R}$ etc.) and use the isomorphisms between Clifford algebras of different signatures ($\mathcal{Cl}(p+1,q+1) \cong \mathcal{Cl}(p,q) \otimes \mathcal{Cl}(1,1)$ etc.) to bootstrap the equivalent matrix algebra isomorphisms of Clifford algebras (and similarly for their even subalgebras) of arbitrary signature and from there one can see when real forms exist.
Best Answer
A representation of the Clifford algebra could be obtained by creation and annihilation fermionic modes $c_k$ and $c_k^{\dagger}$, by the following definition: $$ \Gamma_{2k-1}=c_k+c_k^{\dagger}\,\,; \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Gamma_{2k}=i(c_k-c_k^{\dagger}) $$
For $k$ being an integer between $0 <k< (d+1)/2$. For directions $\mu$ with negative signature you should put a imaginary unit $i$ up front. The Lorentz generator of
$$ \delta [...]^{\rho}=\frac{1}{2}\theta_{\mu\nu}(\eta^{\rho\nu}[...]^{\mu}) $$
where $\theta_{\mu\nu}=-\theta_{\nu\mu}$ is given by:
$$ \Sigma(\theta)=-\frac{i}{2}\theta_{\mu\nu}\Gamma^{\mu}\Gamma^{\nu} $$
Note that for "rotations" in $(2k,2k-1)$ plane is given by
$$ \Sigma(\theta)=-\frac{i}{2}\theta_{2k,2k-1}(c_k+c_k^{\dagger})(i)(c_k-c_k^{\dagger})=\frac{1}{2}\theta_{2k,2k-1}(-c_kc_k^{\dagger}+c_k^{\dagger}c_k)= $$ $$ =\theta_{2k,2k-1}\left(c_k^{\dagger}c_k-\frac{1}{2}\right)=\theta_{2k,2k-1}\left(n_k-\frac{1}{2}\right) $$
This provides the interpretation of the modes. The modes are the eigenvalues the $\Sigma^{2k,2k-1}$ generator, being $+1/2$ for $n_k=1$ and $-1/2$ for $n_k=0$. The same as $J_3$ for the $d=3$ case.
For even dimensions, there is a chiral matrix commutes with the Lorentz generators (e.g. $\gamma_5$ for $d=4$) and can be written in terms of the modes as:
$$ \bar{\Gamma}=\prod_k(1-2n_k) $$
This means that the eigenvalues of $\bar{\Gamma}$, the chirality, is $+1$ if there is a even number of occupied modes and is $-1$ for odd. Since this matrix commutes with Lorentz generator, for even dimensions we may split a Dirac spinor into two spinors, a chiral spinor ($\bar\Gamma =+1$) and anti-chiral spinor ($\bar\Gamma =-1$).
For odd dimensions one of this gamma matrices will be left over. For the last $k$-mode we take just the $\Gamma_{2k-1}$ and the $\Gamma_{2k}$ is unused. You may note that now there are two matrices that commutes with Lorentz generator, the $\Gamma_{2k}$ and the $\bar\Gamma$. It is important to note that they not commute with each other, so we can use just one of them to split the representation but not both. Whatever matrix we use we will end up with a reduction of the number of modes by one. If there is $n$ modes for $d=2n$, for $d=2n-1$ we have $n-1$ modes.
There is also other way to obtain the odd dimension gamma matrices for $d=2n-1$ by starting with $d=2(n-1)$ and using the $\bar\Gamma$ matrix as the $\Gamma^{d}$. This have $n-1$ modes from the start.
Now you can see that there is a distinction between odd and even dimensions. So, is always expected to have this jumps on the numbers of components when you reduce an even dimension $d$ to $d-1$, or increase an odd dimension $d$ to $d+1$, because of the increasing number of $c_k$ modes.