A journal with high impact accepted my paper after peer review. In the copy-editing phase I noticed some errors -- mostly mixed up references. Can the paper now be rejected due to these errors? What about if I want to update some references?
Formally, a paper can be rejected at any time, even after printing. E.g. IEEE describes such cases, which are mostly plagiarism-related. In your case of mixed-up references, a rejection is unlikely.
Updating references should be a non-issue in the early copyediting phase but problematic in the late stages. You might have to live with published wrong references.
If you ask for too many changes on the page-proofs it might be that you are charged a fee for this. Various journals say they will charge for too many changes as it increases the production cost; but I have never heard of a situation where it actually happened.
Mixed-up reference, especially if these are clerical errors, does not sound that dramatic a change, so this should be a non-issue.
Updating reference is perfectly fine; journals will even actively ask for more complete data if available in the page-proofs if the original manuscript only quoted a preprint for example.
Generally, once you are in the copy-editing phase the scientific decision to publish your paper was already taken. A reason for it not to be publish typically could only be procedural (e.g., you do not agree to sign the copyright form or otherwise mess up the process completely) or possibly very severe problems with the paper that emerge.