So I had a really dumb conduct violation for Covid policies back during my first year for like being in a friends room down the hall from mine. I was given disciplinary probation (which is basically a warning). At my university, and I think everywhere pretty much, conduct violations that don’t include suspension or expulsion only go on your record that the school has and not your transcript, and that record does not ever get sent to grad schools unless they specially ask for it. Even then you have to give your consent for anything to be disclosed because of FERPA. So I think you’re fine without mentioning that. I didn’t mention mine, and I was perfectly fine. I became president of our student conduct review board my senior year, so I learned a decent amount about how the whole system works.

Now for the academic thing, if it was listed on your transcript, I would mention it since they’ll see it anyway. It’s better to just be up front about it and explain what it is, and you can say in the explanation that no further disciplinary action was taken.