A senior colleague that belittles my contributions, expertise and won't acknowledge a published error that I identified
When we went to publish a paper ... my boss made the suggestion that we equally share first-authorship, Emily was so upset that she wasn't the sole first author that she literally screamed in my PI's office about it and then didn't come into lab for 3 days after.


If possible, it is always a good idea to negotiate contributions and authorship of a paper prior to conducting the relevant research and drafting of the paper.  This kind of dispute can best be avoided by having all the researchers sit down prior to the research project and agree on the roles of each of the researchers, and how the authorship will be allocated (i.e., who is a co-author vs an acknowledgement, order of authors, etc.).  In some cases the actual contributions may end up differing from the initial plan, and so a renegotiation of the authorship could occur, but in many cases you can avoid disputes by having a clear set of expectations at the start of the project.  Once a dispute arises, there is going to have to be some negotiation and decision made about the dispute, but in the future it would be best to avoid these problems by agreeing on authorship details in advance.


  ... I'm currently working on several papers that build on those published findings.  In working on these new projects, I've discovered an important error that Emily made in the published paper that we are co-first author on.  She denies that there is a mistake...  The mistake needs to be corrected in order for me to continue with publishing my recent projects (but fortunately does not change the main takeaway of the paper).  Emily plans on telling our PI that I’m just a trainee and that I don't know what I'm talking about and that she did not make a mistake (again, I can definitively prove that she did with the email chain).  She also has some clout with our PI because she's been there much longer than I have and is more experienced.


Firstly, you should do everything possible to see if you, Emily, and your PI, can reach an agreement on whether or not there is an error in the published paper.  I would suggest making a meeting with your PI where all of you can look over the details of the claimed error, check your working and discuss the matter, and see if you can come to agreement about whether or not there is actually an error.  If you can get your co-authors to agree with your view, then you could all submit an erratum to the previous paper correcting the error.  If your co-authors can convince you that there is no error, the matter is also resolved.

It sounds like this might not be possible, and so if you try all this and exhaust all possibility of agreement, then here is what I would suggest.  You are the author of your new papers, so it is up to you what to write in them.  It is perfectly acceptable for you to claim in your new papers that you have identified an error in the previous published paper.  For full disclosure to your reader, you should make your claim in the paper, but also note that the claim of the error is disputed by your co-author.  (You can give relevant details in a footnote, but keep it succinct and professional.  Do not bog the reader down with excessive detail or a big shit-fight.)  Re-run the previous analysis without the error, note the new results, and proceed with your new papers on that basis.  Emily will be just as entitled to submit new papers claiming that she is correct and there is no error.  (Both sets of papers would be vetted by referees, but it might be difficult for them to tell who is right, so you might find that some referees will be reluctant to recommend publication.)

If you are unable to come to an agreement on the alleged error with your co-authors, ultimately that is going to mean that you end up claiming one thing and she ends up claiming another.  If these are your papers, and you have tried and failed to reach agreement on the matter, I see no reason why this issue would require any further intervention of your PI, so her greater "clout" with the PI should not make any difference.  You are entitled to submit research to journals making claims you believe in good-faith to be true.  (If you want to, it would also be legitimate for your to submit an erratum for the previous paper to the original journal, notifying your co-authors, and noting clearly in your erratum that your claim of an error is disputed by your co-authors.  I suspect that the journal might not accept this, but you could submit it if you wanted to.)


  Unfortunately, I lost my cool a little bit and some of my responses to her were harsh and less-than-professional. ...  we both made some statements that were unprofessional.


That is a problem, and you are both going to need to cool it.  (And it is probably worth apologising for any comments that were unprofessional.)  Keep your focus on the work, and the correctness or incorrectness of the methods and results.  If you cannot engage with Emily without making unprofessional comments, you should disengage communication with her.