Academic dishonesty: If there is a section of the application dedicated to adressing this, then devote several sentences, as Tildy describes, to briefly explaining the incident then describing the outcome and what you have learned/how you have changed. If you are able to address this completely in the appropriate section of the app, then I would advise against revisiting this topic in your personal statement. The PS is a chance for you to sell yourself, shine on paper, talk about your goals. It shouldn't be the place to remind others of your shortcomings.
I would say the same for nearly any other negative/controversial situation. Address it in the portion of the app dedicated to that topic, and ignor it in your PS. Remember that at many schools, interviewers only get a limited peek at your app. Some only get the PS and some demographic info. Thus, if you get past the Selection Committee screening, your interviewer will be oblivious to these transgressions uinless you choose to divulge them in your essay or the flow of the interview.
Jobs: If you are motivated to become a physician, the best thing you can do is work with physicians/researchers. You'll get experience, letters of recommendation, and a great insight into the medical profession. There are many jobs at a hospital that you can be trained for: EMT, Respiratory, Anesthesia tech. Also think research, but I think it may be a touch more suspicious to work with a researcher if you have a history of scholarly dishonesty. If you can apply for and get the job without discussing that facet, though, go for it. It will show you are a trustworthy member of a research team.
. So, a short but clear, non-defensive sounding paragraph should do the trick here.
