Sorta' depends on the situation. Normally I'd say greeting/text/sign-off is sufficient -- as you note, address/return address and date are covered by the program -- but depending on who it's from and who it's going to more or less may be necessary. Signatures seem to be a thing used in email only by folks in some sort of official position, ferex, and then often only when acting in official capacity (so you'll see one on, say, a legal notice or some sort of financial transaction or whatev', but possibly not something in-house.).
For what it's worth, this is what I've been using to first-time contact my teachers with questions this semester. It's a little muddly in places due mostly to local dialect ("I've got", ferex.), but more or less what I'd recommend for anything not
hyper-formal:
Good day,
If I've got the email address correct, I'm contacting <Name>, instructor for the online <Course name> <Course number>. This is <MYNAME>, a student for the upcoming summer semester. I'm emailing you in order to inquire as to the extent the class will require on-campus meetings or tests. As I'm taking a few of other online classes (specifically <course numbers>) this upcoming semester as well as an on-campus class (<course number>), I'm worried about potential schedule conflicts that may arise between these classes and whether it will be possible to make arrangements ahead of time to compensate for any overlaps. Any information you can provide would be much appreciated.
Thank you for your time,
<MYNAME>
Shakeninja link is good ninjalink.