sometimes you can see if it is singular or plural only by article or adjective in front of the noun or by following verb.. you have to know particular noun. in this case. singular is "der Lehrer trinkt" and plural is "die Lehrer trinken"
In front of Lehrer, yes, but that is not always true. Such as Die Mutter (the mother) in plural is Die Mütter. Have to be careful
Not all teachers are professors (and not all professors, at least in the US, have teaching as part of their responsibilities;) "teacher" and "professor" are two distinctly different concepts.
Yeah you're right! The ending "-in" for an occupation etc. is a typical female ending: Lehrer / Lehrerin (teacher), Arzt / Ärztin (doctor), Bauer / Bäuerin (farmer), Spieler / Spielerin (gamer) and so on