Is there a "statute of limitations" on plagiarism? After being accused in one course, can my work in past courses be re-examined? I believe I unknowingly plagiarized in a past course.
Actually, anyone reading your work can accuse you of plagiarism if it is noticed that you copy the ideas of others without attribution.
Ordering an investigation, however, is a different matter and would depend on local rules and customs. But universities like to protect their own reputation and so some "due diligence" is required. This would be especially true for graduate degrees, I'd expect.
But once you have been accused and not exonerated, then it would be natural for people to look for other instances. In a serious case, such as a thesis, you would bear considerable risk.
Edited to add: No, there isn't a time limit or statute of limitations. No statutes at all in most places. It is what it is unless corrected. If the work can be seen you can be accused. This is distinct, of course, from local disciplinary rules.
Anyone might accuse you at any time, even decades or centuries past your death. However, the burden of the proof is to the accuser.
Accusing someone of plagiarism is a serious issue, and normally, the people who does the accusation have proofs sustaining the accusations.
Once accused, you are expected to defend from the accusation. If you are actually guilty, it is likely that there is nothing you can do to defend yourself and the best you can do is to apologize and recognize your error. However, the academia tends to be unforgiving and ban you forever from working on anything seriously even if you actually apologize.
If you are innocent, defending from a plagiarism accusation is hard. There are a lot of cases where innocent people had no real chance to defend themselves from unfair accusations of this type. What you should try to do is to prove that there was no plagiarism at all and that you properly quoted everyone you should quote, which is a very tiring, stressful and cumbersome process.
Sometimes, someone else plagiarized your work and the accuser thought it was the other way around. In this case, you should accuse that someone else as part of your defense.
Sometimes a defendant defends by counter-attacking and plays the "uno reversal card": The accuser is actually who is plagiarizing work! If the defendant is innocent, this is probably the case, but proofing it might be hard. If the defendant is guilty, this would be part of their strategy to steal other people work.
In cases of a work with several authors that is accused of plagiarism, this might create a serious issue as possibly, only one of the authors did the plagiarism and the others were simply trusting him. If the other authors aren't able to clear themselves, everyone would be considered guilty.
It seems that you previously plagiarized stuff as part of a work done in the context of a previous discipline. If you are a grad student and plagiarized due to not actually knowing how to properly quote stuff or as a result of a very sloppy work, the teacher, while still giving an F or a zero to your work, might then just choose to rub that under the rug. I once caught a student who made exactly that and what I told him was simply "quote all those things correctly and resubmit your work as soon as possible, otherwise there is no other grade than a zero", I also gave him a 10-minute informal lecture about how to properly quote stuff. He did fixed his work and resubmitted, which is a much better outcome than just ruining his academic life.
I believe that the process varies widely by institution and country. In my experience working for a public university in the United States, I needed to file a report for each student accused of plagiarism with the university. I need to provide documentation of plagiarism, an explanation of why this constituted plagiarism, and course materials about the assignment in question. After this process, the university would contact the student to see if they wanted to dispute the accusation or accept a failing grade, with a requirement to take a remedial course regarding plagiarism in order to graduate. This was a thorough, time-consuming process so I did not do it unless it was a clear violation of plagiarism. For students who seemed to have issues with citations, for instance, I would give them a warning that they needed to follow proper rules or else they would be reported to the university. Again, this process can vary widely, but in my instance as a professor, I only reported plagiarism that occurred within the classes I taught while the classes were in session. Unfortunately, I usually report two to three students for plagiarism each year.
There is certainly no accepted “status of limitations”. Researchers now have old papers retracted because plagiarism is discovered by other unrelated academics sometimes years after the fact.
As an example a German rabbi has recently been found to have plagiarized part of his 1992 dissertation. This is not a university example but you can find a large number of miscellaneous examples by searching for the keyword “plagiarism” on the Retraction Watch website and find those related to university, including some cases where degrees were withdrawn, as this case of a US Senator.
So: it does happen that people look back on old work, and it does happen that degrees are revoked as a result of plagiarism.
There are no real rules, but as you allude, things can get fairly complex, as there are two fairly conflicting goals of Academic Honesty procedures in US schools.
One goal is to defend the value of the diploma from a given school. If employers figure out that a school graduates a large number of cheaters, they will (appropriately??) weight the value of a diploma from that school in a downward direction. Toward that goal, finding every instance of dishonesty in a student's history at that school may be a valuable exercise.
The other goal would be to modify the behavior of a student caught cheating, providing the opportunity of learning from the experience and learning how not to cheat. With that in mind, if one delves too deeply into past behavior, one might find a veritable well of plagiarism, bringing the level of penalty up to permanent separation, thus removing the pathway of modifying behavior.
These goals may well conflict. The ideal medium would be to identify cheating every time and immediately upon the event, but that really can't happen.
Practically, it would be difficult to gather all of a student's past (and passed!) assignments and put them under a microscope. It's also a big ask for faculty members.
Note that for faculty caught plagiarizing, it's fairly likely that their life work will be put under a microscope, that that's probably appropriate.
In Germany in 2013, 33 years after writing her doctoral thesis, the education minister Annette Schavan was stripped of here doctorate by her Alma Mater, Heinrich Heine University in Düsseldorf. She subsequently resigned her post. Your country may differ but educational institutions and employers can have long memories in this regard.
At my institution each department has a plagiarism officer, who deals with cases reported by faculty according to the procedures set out in university regulations. I am aware of at least one case where the plagiarism officer conducted a review of a students previous work because of the nature of their offense (after it had been investigated and punished appropriately). So retrospective investigations can and do happen, if the institutions regulations allow it (and there is good reason).