That is a very good question; I've never thought about it before, and I don't know the answer. You can say "tomorrow evening" or "yesterday evening"; you can say "today, in the evening"; you can say "today's evening" when talking about an event in the evening; but you cannot say "today evening".
Today - may refer to any portion of the day: morning, afternoon, or evening.
e.g. "I am going to the dentist today." (could be any time of day)
Evening - denotes some time late in the day.
e.g. "I am going shopping this evening."
It doesn't make sense grammatically or logically to include both 'today' 'evening' in the same clause.
It's not a question of logic, as Pfiff's answer demonstrates (with "yesterday evening" and "tomorrow evening", and don't forget that any name of a day of the week can be used with "evening" as well). It's simply a question of custom. We don't say it because we don't say it.
It may be influenced by the fact that "this evening" is commonly understood to be today's evening in English, so we have a meaningful phrase that works, and that pushes out the "today evening" alternative, whereas with the phrases incorporating the other days we have no meaningful substitute to take their place.
"Today", whilst denoting the whole day also has strong connotations of being day time (when the sun is up). Without any information other than "today", it is accepted that you are referring to the day time. So the term "today evening" doesn't go together well. Similarly, it is also accepted that unless it has been stated otherwise, the day in question is always the present day. So, "today evening" also sounds redundant. Just using "evening" would have been taken to mean that you are referring to the evening of this day.
"Today I'm going to the dentist" - during daylight.
"In the evening we're going dancing" - present day.
"I'm working tonight" - during the night, typically present day, however "tonight" can also refer to the whole blanket of night until sunrise.
I don't think so.
Where it could translate as "now", I believe the sense would be more like "nowadays", which wouldn't work in your sentence along with "tonight".
Your other problem with respect to this sentence would be that if "heute" were "now", you would be missing a word in the German sentence to clarify that it was "this evening" (i.e. "tonight" in your sentence) that you were talking about.
No. If you're being literal, then "I work today evening".
There's nothing in here that would correspond to the possessive 's.
Also, the fact that heute is lowercase shows that it's considered an adverb -- and those don't have possessive endings anyway. i.e. it's not the noun "a today" but just the adverb "today".
Duolingo is a stupid computer. It can only match answers against lists of acceptable translations. It doesn't actually understand English or German.
So if an answer doesn't appear on the list, it will get rejected.
The lists can't contain all possible non-native-speaker mistakes -- that would make them enormously huge.
If you need leniency, there's no substitute for a real human teacher who can actually think.