What is the value $P(x)$ supposed to be?
Nothing. We are interested in the coefficients of the polynomial $P(x)$, which are restricted to the Booleans $\{0,1\}$. These coefficients reflect the wiring of the LFSR. For other polynomials, they could reflect the state of the LFSR. We seldom¹ need to evaluate that polynomial $P(x)$, or other polynomials we use, for a particular value of $x$, or even specify the set $x$ belongs to. Think of $x$ as an unspecified variable, be it in integers $\mathbb Z$, rationals $\mathbb Q$, reals $\mathbb R$, complexes $\mathbb C$, as you see fit; and confidently perform arithmetic on such polynomials per the standard rules of algebra, modified by $1+1=0$ (equivalently, by reducing all coefficients of polynomials modulo $2$).
The far right flip flop is represented by $1$. Is this shorthand for $x^0$?
Yes. Another reason we write $1$ is to avoid having to define $0^0$.
The four flip-flops are labelled $0,1,2,3$. So why a power-four term?
The degree-four term is only in $P(x)$, which represents the wiring of the LFSR, not the state of it's flip-flops. When dealing with the state, it will be represented by a polynomial $S(x)$ of degree at most three.
Also: when we reduce any polynomial modulo the polynomial $P(x)$, the remainder $S(x)$ has degree strictly lower that $P(x)$, thus it's coefficients (typically a new state of the LFSR) fit the four flip-flops.
Yet another way to see it is that the term $x^4$ in $P(x)$ corresponds to the bit that gets out of the shift register when we shift by one bit (equivalently, multiply the state by $x$), while the other bits correspond to adjustments of the new states of each flip-flop.
Makes me wonder if this 'equation' isn't really an equation you expect to emit a value to use, but some kind of hand-wavy way to just describe the architecture of a LFSR?
Indeed, $P(x)$ is about the architecture of the LFSR. The representation as polynomial $P(x)$ for architecture and $S(x)$ for state is useful to establish properties of LFSRs. In particular, for a LFSR in Galois form, the state evolves per $S_{i+1}(x)=S_i(x)\,x\bmod P(x)$, from which it follows $S_i(x)=S_0(x)\,x^i\bmod P(x)$.
Note: here, $\bmod$ yields the remainder per polynomial division, again with coefficients in the Booleans.
¹ Exceptions: evaluation of $P(x)$ at $x=1$ in the Booleans, or $x=2$ for the integers, yield occasionally interesting values.
