Let $F$ be a field, and let $f_1,f_2,\ldots, f_k\in R:=F\langle\langle x,y\rangle\rangle$ with $k\in \mathbb{N}$. Order monomials in $R$ by degree, and then lexicographically. Since the question concerns computability, assume that there is an algorithm which spits out the coefficients of the monomials in $f_i$ (in order). I claim that:
Proposition: Given any fixed $n\in \mathbb{N}$, there is an algorithm to decide whether $I^n\subseteq (f_1,f_2,\ldots, f_k)$.
To prove this, first we need a lemma and some notation. Given $r\in R$ we write $r[n]$ for the homogeneous component of $r$ in degree $n$.
Lemma: $I^n\subseteq (f_1,f_2,\ldots, f_k)$ iff for each of the $2^n$ monomials $m$ of total degree $n$ there is a power series $g_m\in (f_1,f_2,\ldots, f_k)$ such that $g_m[n]=m$.
Proof of the lemma: The forward direction is obvious. For the reverse, let $m_1,m_2,\ldots, m_{2^n}$ be the list of all monomials of degree $n$, and let $g_i:=g_{m_i}$. Fix $h\in I^n$. We can write $h[n]=\sum m_i a_{i,n}$ for some $a_{i,n}\in R$ (all of degree $0$). Setting $h':=h-\sum g_i a_{i,n}$, we see that $h'$ has zero homogeneous components in degree $\leq n$. We can write $h'[n+1]=\sum m_i a_{i,n+1}$ for some $a_{i,n+1}\in R$ (all of degree $1$). Set $h''=h-\sum g_i(a_{i,n}+a_{i,n+1})$. Repeating this process, we obtain power series $a_i=\sum_{m\geq n}a_{i,m}$ such that $h=\sum_i g_i a_i\in (f_1,f_2,\ldots, f_k)$.$\qquad \blacksquare$
Proof of the proposition: By the lemma, it suffices to decide for each monomial $m\in I^n$, whether there is a power series $g_m\in (f_1,f_2,\ldots, f_k)$ with $g_m[n]=m$. We may as well work in the quotient ring $R/I^{n+1}$. The image of the ideal $(f_1,f_2,\ldots, f_k)$ modulo $I^{n+1}$ is a finite-dimensional $F$-vector space, and so the question is easily decided by a row-reduction style argument. [More concretely, one can modify the set $\{f_1,f_2,\ldots, f_k\}$ modulo $I^{n+1}$ so that no leading terms are linear combinations of others, and the set is closed under left and right multiplication by $x$ and $y$. Then, the leading terms of degree $n$ either do or do not generate all of the appropriate monomials.]$\qquad \blacksquare$
To give an interesting example, consider $f_1=yx-y^2+x^3$, $f_2=xy-x^2$. One can show directly (using the methodology above) that $I^{5}\subseteq (f_1,f_2)$. If we modify $f_1=yx-y^2+x^k$ (with $k\geq 3$), then we get $I^{k+2}\subseteq (f_1,f_2)$.
On the question of whether it is decidable, given as input only the algorithms describing the power series $f_1,\ldots, f_k$, whether an $n$ exists for which $I^n\subseteq (f_1,f_2,\ldots, f_k)$, I don't know the answer.
Quite a lot of questions here!
It is perhaps worth making a distinction between scalar classical probability theory - the study of scalar classical random variables - and more general classical probability theory, in which one studies more general random objects such as random graphs, random sets, random matrices, etc.. The former has the structure of a commutative algebra in addition to an expectation, which allows one to then form many familiar concepts in probability theory such as moments, variances, correlations, characteristic functions, etc., though in many cases one has to impose some integrability condition on the random variables involved in order to ensure that these concepts are well defined; in particular, it can be technically convenient to restrict attention to bounded random variables in order to avoid all integrability issues. In the more general case, one usually does not have the commutative algebra structure, and (in the case of random variables not taking values in a vector space) one also does not have an expectation structure any more.
My focus in my free probability notes is on scalar random variables (commutative or noncommutative), in which one needs both the algebra structure and the expectation structure in order to define the concepts mentioned above. Neither structure is necessary to define the other, but they enjoy some compatibility conditions (e.g. ${\bf E} X^2 \geq 0$ for any real random variable $X$, in both the commutative and noncommutative settings). In my notes, I also restricted largely to the case of bounded random variables $X \in L^\infty$ for simplicity (or at least with random variables $X \in L^{\infty-}$ in which all moments were finite), but one can certainly study unbounded noncommutative random variables as well, though the theory becomes significantly more delicate (much as the spectral theorem becomes significantly more subtle when working with unbounded operators rather than bounded operators).
When teaching classical probability theory, one usually focuses first on the scalar case, and then perhaps moves on to the general case in more advanced portions of the course. Similarly, noncommutative probability (of which free probability is a subfield) usually focuses first on the case of scalar noncommutative variables, which was the also the focus of my post. For instance, random $n \times n$ matrices, using the normalised expected trace $X \mapsto \frac{1}{n} {\bf E} \mathrm{tr} X$ as the trace structure, would be examples of scalar noncommutative random variables (note that the normalised expected trace of a random matrix is a scalar, not a matrix). It is true that random $n \times n$ matrices, when equipped with the classical expectation ${\bf E}$ instead of the normalised expected trace $\frac{1}{n} {\bf E} \mathrm{tr}$, can also be viewed as classical non-scalar random variables, but this is a rather different structure (note now that the expectation is a matrix rather than a scalar) and should not be confused with the scalar noncommutative probability structure one can place here.
It is certainly possible to consider non-scalar noncommutative random variables, such as a matrix in which the entries are themselves elements of some noncommutative tracial von Neumann algebra (e.g. a matrix of random matrices); see e.g. Section 5 of these slides of Speicher. Similarly, there is certainly literature on free point processes (see e.g. this paper), noncommutative white noise (see e.g. this paper), etc., but these are rather advanced topics and beyond the scope of the scalar noncommutative probability theory discussed in my notes. I would not recommend trying to think about these objects until one is completely comfortable conceptually both with non-scalar classical random variables and with scalar noncommutative random variables, as one is likely to become rather confused otherwise when dealing with them. (This is analogous to how one should not attempt to study quantum field theory until one is completely comfortable conceptually both with classical field theory and with the quantum theory of particles. Much as one should not conflate the superficially similar notions of a classical field and a quantum wave function, one should also not conflate the superficially similar notions of a non-scalar classical random variable and a scalar noncommutative random variable.)
Regarding localisable measurable spaces: all standard probability spaces generate localisable measurable spaces. Technically, it is true that there do exist some pathological probability spaces whose corresponding measurable spaces are not localisable; however the vast majority of probability theory can be conducted on standard probability spaces, and there are some technical advantages to doing so, particularly when it comes to studying conditional expectations with respect to continuous random variables or continuous $\sigma$-algebras.
Best Answer
One common way to quantify non-commutativity which is especially popular in operator algebra theory and non-commutative geometry is to use the Schatten norms. Given a bounded operator $T$ on a separable Hilbert space $H$ one defines the Schatten $p$-norm $||T||_p$ (for $p \geq 1$) of $T$ to be the trace of the operator $|T|^p$ defined using the functional calculus. More explicitly, if the operator $\sqrt{T^*T}$ has countable spectrum (a necessary condition for $||T||_p$ to be finite) then
$$||T||_p^p = \sum_n \lambda_n^p$$
where the sum is taken over the spectrum of $\sqrt{T^*T}$. In particular the case $p = 1$ corresponds to the trace norm and $p = 2$ corresponds to the Hilbert-Schmidt norm.
In the finite dimensional case the Schatten norms are of course all finite, but in the infinite dimensional case it is useful to measure the non-commutativity of two operators $A$ and $B$ by calculating the minimum value of $p$ such that $||[A,B]||_p < \infty$. For example one can measure the regularity of a function on a manifold by asking which Schatten classes its commutators with appropriately chosen (pseudo)differential operators belong to. This is the basis of Connes' notion of a non-commutative manifold.
Also note that every Schatten class operator is compact, and for many purposes it is useful to replace the condition that two operators commute with the condition that their commutator is compact. Of course this particular notion has no finite dimensional analogue.