When writing a technical article that includes mathematical proofs: Is it acceptable to have footnotes within the body of a formal proof (to elaborate/clarify non-essential points) or is the use of footnotes in proofs considered bad practice?
Like almost anything else in writing, style should follow function. In other words, if you think that a footnote makes it easier for your reader to follow a text (for example, because it explains an aside that is too long for a parenthetical remark), then it is appropriate to use one. There is no general guideline whether footnotes are acceptable or not. It all depends on what you want to say and how you want to say it -- choose whatever means you think are appropriate to tell the story of the proof.
I personally find footnotes exactly as acceptable or unacceptable in a proof as in other parts of a paper. A well-written proof often contains quite a lot of explanatory prose, and there is nothing special or magical about a proof that prevents one from having an "aside" comment within it.
I would, however, find it very strange to have a footnote in the statement of a theorem, just as I would find it strange to have a footnote in an equation.
I cannot remember ever having seen a footnote in a proof. My instinct would be to put elaborations or clarifications into a remark after the proof.
In an actual math paper, you could actually have a "Proposition 1", followed by its "Proof", then a "Remark 2". For instance:
Remark 2. Note that $X$ in the preceding proof does not have property $y$, which would have allowed us to use the technique from Foo & Bar (2015).
However, this may well depend on your field, on your journal, on the editor and on the referees. Some of these may well frown on footnotes, while others may be fine with them.
I'd say you'd be safest with putting additional material into Remarks.
Footnotes within proofs are fine and to be recommended.
I've read tons of (multipage) proofs (in signal-processing, or machine learning) where the author's train of thought was impossible to follow and obscured by unnecessary steps and diversions. Twenty pages of matrix algebra and higher-order derivatives in order to establish some underwhelming pseudo-result which follows directly from well-known basic theorems familiar to people in the field; or else can be sketched out in one or two paragraph of paraphrase, to articulate the necessary motivation before wading into verbose proofs.
A proof [in most academic domains] should be aggressively made as compact as it can be, without removing anything essential. Shunt all non-essentials, sidebars and footnotes into footnotes or remarks.
One frequent paradigm (common in signal processing): assume an independent normal distribution on multiple variables (where this is known to be a ridiculously invalid assumption), waste 8+ pages deriving a non-result based on that assumption, before instantly discarding and invalidating that result and introducing a new (but much weaker) derivation with the proper assumptions that should have been used in the first place, but render the subsequent result very basic and uninformative. Finally fall back on showing a few graphs or experimental results to discuss how the process behaves in reality .