(Localized answer: I'm in Germany and in my work experience also [the north of?] Italy thinks along the same lines. Both languages have the concept of changing pronouns as well as first name vs. last name to express different grades of formality and closeness)
Spoken interaction with students
The default here is to address TAs by "Sie" (formal pronoun) and last name. Addressing in everyday spoken language by Dr. is becoming more and more unusual, it is still the default with people who are much older and/or are known to put much emphasis on this formality. Same with Prof., the default being slightly more on formal side. Needless to say, the TA or prof also adresses the students by "Sie" and last name.
For mass courses (labwork practica where I have different groups of students on every occasion or seminars), we just stay with the formal way of adressing. These courses typically have a comparably low number of one-to-one interactions with the student.
When students join our research group, I offer the "Du" (informal pronoun): this is done by (re)introducing yourself "By the way, I'm Firstname".
As a student I found it very awkward if the teachers weren't clear about this: in our culture it is clearly up to the more senior (also or even mainly professionally more senior) person to offer to drop the formalities.
For inner-German emails I'd still consider it rude not to put an opening line and a closing line to the body. For e-mail exchange with other countries I adapt to their customs as far as I know them. In my language the way the recipient is adressed and the email is signed state how formal or close the relaionship is to be. This is information the email adresses and the full email signature cannot provide. The full signature below the "--" line is the place where full professional grades and position go.
Emailing with students without these "instructions" may be perceived as rude or also as insecure. The mass-course email starts with "Dear Mr./Ms. X", or less formally "Dear seminar group A" and ends with "Best, Firstname Lastname" or less formally just with "Firstname Lastname".
I close with abbreviation ("VG C") only with close collaborators. In that case, opening and closing line may be dropped as well.
Students becoming too friendly
I'll try to live up to the stereotype that Germany are direct to the level of being rude. Here are my thoughts:
In German language, the concept of too friendly with a negative connotation does not exist, friendly is unambiguously positive. From that perspective, I'd say that your "too friendly" is a euphemism for something along the lines of presuming and rude, not respecting you.
Now in the described situation I'd try hard to avoid any euphemisms about the student behaviour as they may be perceived as a sign of you lacking confidence in yourself, and submitting to the badly behaving student: your language offers them a very easy way to ignore your request. If that happens (and I'd think it more likely to happen with rude students...), good-bye to the student respecting you.
Even (or maybe: particularly?) in a culture that relies less on formal distinctions (like the formal way of adressing) if someone doesn't know and doesn't get the hints how to behave themselves, it may help to tell them in clear words what is expected and that not behaving accordingly leaves a very bad impression. I'd take them aside to tell that, and I'd make a point that I don't particularly grudge this first time - but that I'm concerned because in a professional environment such non-respecting behaviour may cut their throats.