Have any complexity classes been shown to be equal or non equal in the past five years?
This is probably better as a comment, but too long.
Gordeev and Hauesler put forth a proof claiming NP = PSPACE a couple months ago. While most complexity theorists believe NP $\neq$ PSPACE, no flaws have been found in this paper. The authors have also responded to potential critiques, indicating how they avoided the expected pitfalls (https://www.cs.nyu.edu/pipermail/fom/2016-October/thread.html#20130). In my opinion, it sets them apart from most of the folks who submit papers (falsely) claiming something along the lines of P = NP (which are a dime a dozen).
I have not gone through the paper closely and am not versed in logic well enough to have an opinion on its correctness. So I am not endorsing it. Also, this paper has not been widely accepted by the complexity theory community. If it is true, this would qualify as an answer to your question, though (and a major result- it implies that NP = co-NP and that the polynomial hierarchy collapses).
Their paper: https://arxiv.org/pdf/1609.09562v1.pdf
A worthwhile reddit thread, where folks with more of a complexity background than myself weigh in- https://www.reddit.com/r/compsci/comments/56w8hy/arxiv_paper_claiming_np_pspace_any_immediate_red/#bottom-comments
In [1], the authors show (among a bunch of other things) that $TC^1 = AC^1[p_n]$ where $p_n$ is the $n^\text{th}$ prime and $n$ is the number of input bits to the circuit.
Depending on how strictly you read the original question, this may or may not be a satisfactory answer. First, the Complexity Zoo is quite good for what it is but is far from comprehensive: it includes a definition of $AC^1$ under its entry for $AC$ (via definitions of $AC^i$) but offers no such parallel definition for $TC^1$ despite it being a perfectly standard complexity class. Also, it has no entry for $AC^i[m]$ even though it's reasonably easy to infer from the surrounding context for $AC^i$ and $AC^0[m]$. Second, the classical view of $AC^i[m]$ is that the modulus is constant with respect to the size of the input; the definition given in the Complexity Zoo does not require this.
- Allender, Eric, Anna Gál, and Ian Mertz. "Dual VP classes." International Symposium on Mathematical Foundations of Computer Science. Springer Berlin Heidelberg, 2015.