Yes, "trade" should also be accepted. If it was not, you should report it. http://dictionnaire.reverso.net/allemand-anglais/handeln
That meaning would require more information (deal what?). http://dictionnaire.reverso.net/allemand-anglais/handeln
Because, as far as I know, the English '(to) deal' doesn't quite work like that. I would interpret your sentence to either mean that you will not deal drugs (in German '(mit Drogen) dealen' or 'mit Drogen handeln') or that you won't deal out (playing) cards ('Karten austeilen'). Now, '(to) deal in sth.' can be translated to 'mit etw. handeln', but to my knowledge it needs the preposition, i.e. you might say "I will not deal in anything." as a translation for German "Ich werde mit nichts handeln.", but I don't think that your sentence on its own would be understood in the same way (unless some native speaker wants to correct me).
" I will not act" must surely be an acceptable translation?
"I will not act" is an accepted translation.
Do you have a screenshot of that sentence being rejected in a translation exercise?
If so, please share it with us -- upload it to a website somewhere (e.g. imgur) and tell us the URL of the image.
Some conservative English speakers use “you will, he will, they will” but “I shall, we shall” for a simple future meaning. For those people, ich werde might be better translated as “I shall”.
This course, though, uses “will” for the simple future regardless of subject, which is how many native speakers use English nowadays. Thus, we do not consider “shall” better; the default/best translation will have “will” even if the subject is “I” or “we”.
If you wish to use “I shall”, it should generally be accepted as an alternative.