'He is a teacher and I am also." is not a correct way to translate this?
The problem stems from which translation you chose to use. While "aussi" can mean "also", it doesn't mean that in this situation. In this situation, you would use the "too" translation, since that would be the correct way to write it in English.
On a side note, Duolingo also accepts "as well" for "aussi", because "too" and "as well" are synonymous in English.
"... and I as well" is the most gramatically correct version but "... and me too" is idiomatically correct too.
In French, when you say what someone's job is, there is no un or une. e.g Il est professeur, Elle est avocate.
Professions act like adjectives. They are considered an attribute of the person. It's like "I am an Australian". You can say "I am Australian" as well.
While it is correct, it isn't proper French, the correct way would be moi aussi.
Why are the articles dropped for occupations when être is used?
we would say: he is a professor.... not: he is professor
I noticed in an earlier question, 'des' was included before soldats;
Ce sont des soldats (as in a particular group of not in general?) but not included in - Ils sont soldats.
Is there a reason for why un, une or les or whatever absent most times and why would it be correct as in the above example?
Ce sont des soldats - These/those are some soldiers
Ils sont soldats - They are soldiers
Professions are usually treated as adjectives although I've been told that using "un(e)" is not wrong, it's just not how the French say it.