(1) How `fine' a tensor decomposition can we obtain of $H$?
Without any additional constraints, the answer is: as fine as you want.
For example, it can be factorized as
$$
H = H_x\otimes H_y\otimes H_z\otimes H_\text{spin}
\tag{1}
$$
where ${\cal H}_x$ corresponds to the particle's freedom of motion in the $x$ direction, and similarly for the $y$ and $z$ factors. In this decomposition, the spin factor $H_\text{spin}$ is finite-dimensional (dimension $2j+1$ for a particle of spin $j$), and the other factors are infinite-dimensional. Beware that the word "dimension" will be used here for two different things: the number of dimensions of space, and the number of dimensions of the Hilbert space. The latter number is infinite.
To see that the decomposition (1) is legitimate, use the fact that any single-particle wavefunction may be written $\psi(x,y,z,m)$, where $x,y,z$ are real variables and $m\in\{1,2,...,2j+1\}$ is a discrete variable, and then use the fact that any such function may be expressed as a (continuous) linear combination of factorized functions $\alpha(x)\beta(y)\gamma(z)\mu(m)$. The factorization (1) is "obvious" in the sense that we implicitly already have that factorization in mind any time we write the wavefunction as a function of $x,y,z,m$.
But the same Hilbert space $H$ can also be written as a product of arbitrarily many factors. The key to understanding this is to remember two things:
$H$ is an infinite-dimensional separable Hilbert space, which means that it contains a countable set of basis vectors whose linear combinations can approximate any other state-vector arbitrarily well. An example of a countable basis for $H_x$ is the set of wavefunctions of the form $x^n\exp(-x^2)$ for $n\in\{0,1,2,3,...\}$.
All infinite-dimensional separable Hilbert spaces are isomorphic to each other, meaning that they are identical as Hilbert spaces even though we may write them in very different-looking ways. For example, $H$ is isomorphic to $H_x$.
If this unique infinite-dimensional separable Hilbert space is denoted $H_\infty$, then these two facts imply the following isomorphisms:
\begin{align}
H &= H_\infty \\
H &= H_\infty\otimes H_\infty \\
H &= H_\infty\otimes H_\infty\otimes H_\infty \\
H &= H_\infty\otimes H_\infty\otimes H_\infty\otimes H_\infty \\
H &= H_\infty\otimes H_\infty\otimes H_\infty\otimes H_\infty\otimes H_\infty
\end{align}
and so on. For example, the Hilbert space of a single particle in $3$-dimensional space is isomorphic to the Hilbert space of a single particle in $N$-dimensional space, for arbitrarily large $N$. Even better: the Hilbert space of a single particle in $3$-dimensional space is isomorphic to the Hilbert space of quantum chromodynamics, with its rich spectrum of mesons and baryons.
The important message here is that physical systems are not distinguished from each other by their Hilbert spaces. Instead:
- The distinction between different physical systems in quantum theory is encoded in the association between specific physical observables and specific linear operators on the Hilbert space.
The difference between a single-particle system in $3$-dimensional space and a single-particle system in $42$-dimensional space has nothing to do with their Hilbert spaces; it has everything to do with which operators we designate as "position operators" (or "momentum operators") along specified spatial dimensions. When we use a function of the form $\psi(x,y,z,m)$ to represent an element of $H$, we are implicitly saying something about which operators we have designated to represent position operators along specified spatial dimensions — and also which operators represent the particle's intrinsic angular momentum.
(2) What are the implications measurement-wise?
This may or may not be answered by the preceding remarks about observables, depending on what kind of "implications" are of interest. When we represent an element of $H$ as a function $\psi(x,y,z,m)$, we are typically implicitly declaring that $-i\hbar\partial/\partial x$ represents the $x$-component of the particle's momentum, and similarly for the $y$ and $z$ components. The fact that these operators commute with each other is a necessary condition for associating them with different factors in some factorization of the Hilbert space. However, just because two operators commute with each other doesn't mean that we can or should associate them with separate factors. For example, in a two-dimensional Hilbert space, we can easily construct two operators that commute with each other; but we can't factorize a 2-d Hilbert space into smaller factors.
(3) Can there ever be entanglement between the constituent tensored Hilbert spaces?
Yes, absolutely. For example, with respect to the factorization (1), almost all wavefunctions $\psi(x,y,z,m)$ are entangled with respect to $x,y,z$. The only ones that aren't entangled are wavefunctions of the form $\psi(x,y,z,m)=\alpha(x)\beta(y)\gamma(z)\mu(m)$. The way the word is most often used, "entangled" just means that the given wavefunction (or state-vector) can't be factorized with respect to the given factorization of the Hilbert space. A wavefunction that is entangled with respect to one factorization may be non-entangled with respect to a different factorization. When we use a particular factorization of the Hilbert space, it's usually because we are implicitly saying something about the observables associated with those factors, and that's what makes entanglement interesting.
The moral of this story is that Hilbert space is a featureless place, devoid of any physical content on its own, and we can factorize an infinite-dimensional Hilbert space in countless different ways, none of which have any physical significance on their own. In quantum theory, the association between physical observables and linear operators is what distinguishes different models from each other.
Of course, a factorization of the Hilbert space can facilitate describing such an association. Statements like "subsystems correspond to Hilbert-space factors" are common in the literature, and that statement is really an abbreviation for this one: "observables represented by linear operators on the $n$th factor are observables associated with the $n$th subsystem." The association bewteen physical observables and linear operators is what matters.
Long answer, simple message.
