The contents in this wiki are written by students based on their personal experiences, so there may be some information that is inaccurate, incomplete, or may become outdated. For critical information, it is advisable to double-check using other reliable sources.
Very important stuff, and thankfully Cornell has a pretty well laid out resource, as well as a very helpful & responsive department for this.
Maintaining your status (general)
https://international.globallearning.cornell.edu/maintaining-your-status
[relevant screenshot]
Things to know when you travel
https://international.globallearning.cornell.edu/travel
Remember that you need a valid travel signature in your I-20 (for F1 visa) or DS-2019 (for J1 visa). It is only valid for 1 year so you have to renew this every year.
Other travel documents that are good to bring just in case are explained in the link above.
Other resources & getting help
International Services at the Office of Global Learning cover a lot of immigration related affairs.
https://international.globallearning.cornell.edu/
You can also get advice from their expert staff by appointment (in-person or Zoom), Email and Phone. You have to schedule appointments in advance (1-2 weeks usually). For emails, from my experience, they respond really fast if your situation is urgent. For phone, sometimes they don't pick up immediately, but I have had an experience when I dropped a voicemail explaining my urgent situation and they called me back fast.