If you want a guideline regarding how the students should address you, go by the culture of the department, if the other professors are using last names, use last names, if they're using first names, use first name, if there's no clear pattern, use whatever is comfortable to you. If you sign your emails "Firstname Lastname," it's not an invitation to do anything, since full names are awkward to use in conversation, and it puts the burden on the other person to decide how to address you.

> I am wondering whether asking my students to call me Dr. LASTNAME would resolve some of the rudeness that I've encountered.

**It won't.** If someone is being inappropriate, react immediately in a stern voice with "Mr/Miss [lastname], that is inappropriate." Using their last name conveys an immediate breaking of the familiarity between the two of you, showing that you mean business.

Regarding **favors** like extensions and flexibility, you need to be clear from the start on what your policy is, **outline the policy in the syllabus, and stick to it**. It's not just a matter of respect, but also a matter of fairness to the students: if the students that aren't playing by the rules are getting their way, the rest of the students are being put at a disadvantage. There will of course be situations where the policy will need to be broken, but those should be extreme (death in the family, student hospitalized/severely ill, natural disaster, global war, etc.) and rare.
Sticking to the policy also applies to office hours, btw. You are obviously free to move your office hours for your own reasons, but being overly available to students outside office hours can be a bad thing if it gets out of control.

Disclaimer: I'm not a professor at the time of this writing.