Asking a question in SE might be considered a breach of anonymity?

I think there's two parts to the question, one about anonymity, one about content.

In terms of anonymity, it's unlikely that your paper would be rejected, at least in my field (which is not math/CS). While papers are submitted anonymously, in reality, reviewers sometimes have a good idea of who the author is. A reviewer can recognize the submission's subject area and approach from other work you've published/presented, or be tipped off by little things like the citation of an unpublished thesis (which likely only the author would know about). If reviewers think they know the author, and there's a conflict of interest, they contact the editor; if they don't see a conflict of interest that would bias them toward or against who they think is the author, they just review it. That's been my experience at least.

In terms of content, it's a bit dicier. I know this isn't something you're asking about, but I figured I'd address it for completeness sake. I assume that you're referring to questions that are fairly small, narrow, and specific, and whose answers don't make up the core of your paper. If so, no problem. But an editor or reviewers might take issue with you using answers you get here as your own, especially if they're central to your argument. You probably know that already, and I've never seen it occur in any blind review I've done, but I feel better including this caveat in my answer to your main question.

It's becoming a common practice to cite answers on SE if they get used in papers. I'd strongly recommend (at least for the second question) that you cite answers you get. There is indeed a risk that this violates anonymity requirements (because you're not supposed to cite material that can reveal your identity), but you can get around that by merely citing "an answer on the stackexchange forums" with "identity withheld to preserve anonymity".

In this way, you make it clear whether the result is yours or not, without breaching rules.

You should carefully check the Creative Commons Attributions Share Alike (CC-BY SA) license agreement under which this site operates. You will find a link at the lower right corner of this page or simply use the link. The license essentially tells you that you can use and distribute the material, even commercially, provided that you make correct attribution and that any material that builds on it is shared under the same license. The license can be waived through permission (see the CC-BY SA page).

So from this perspective you are not doing anything wrong as long as you follow the license. I also do not think any journal, editor or reviewer would hold use of an SE site against you as long as the material can pass review, the license is followed and that the source for your information is provided. Also check Attributing contributions to academic work that occur in Stack Exchange for more information.