Should you always use appropriate letter conventions when emailing a teacher? Usually, when I send an email to a teacher/lecturer, I always start "Dear Mr. Atwood" and end with "Regards/Best wishes (etc.) Leo". If I have to send a followup email after they have replied, I omit the opening and closing sections entirely, and just write my message. Is this considered rude? Would you prefer that a student always lead an email with "Dear" and signed their name?
I do the same. In an ongoing correspondence, where the next email is a reply to the previous one, I usually omit the complete introduction and ending sections. However, there are two cases in which I stick to the full option:


The person I'm writing to keeps their emails formal, so I do the same
A significant amount of time has passed since the last response (e.g. a recent update to a past correspondence)


I always place some kind of salute at the end, e.g. Regards, . The complete end section would include a footer with my contact details and affiliation as well my full name.