下を向く is just looking down, moving/keeping your head towards your feet.
見下ろす is used when you are looking at something that is at lower altitudes than where you are - looking at the scenery from a mountain, looking at the clouds from inside an airplane, looking at the streets below from a building, looking at someone from the second floor etc.
So in your case,...
彼女は殺人の有罪判決が確定した seems fine. I think Duolingo is just being annoying there.
As for the additional で, it's not the verb that warrants it, but the noun. 有罪 doesn't itself refer to a verdict but a person's state of being guilty. If you want to specify what crime the person is guilty of, you should use で.
You shouldn't omit this で in a noun ...
Before we begin, this may be technical, but "verdict" and 「判決」shouldn't be equated, because "verdict" and "judgment" are not interchangeable and shouldn't be confused, especially in Japanese contexts.
A verdict is the finding or decision made by a jury. Japan doesn't have a jury system 「陪審制」, and thus "verdict" doesn't ...
The basic meaning of もの is just thing rather than something. なにか/なんか adds the nuance of "seeking" (i.e., you are trying to find/identify something that fits the criteria).
飲むもの things to drink, beverage
何か飲むもの something to drink
食べたいもの things you want to eat
何か食べたいもの something you want to eat
怖いもの a/the scary being (may be already identified)
The た-form would be correct if the sentence ended there.
The compound verb 勉強してくる is used in its て-form to connect these two sentences into one, in one of the most basic functions of the て-form.
The sentence sounds a bit unnatural to me, though. I would probably use an adversative conjunction.
In most cases you can use them interchangeably.
My feeling is that 日本人女性 sounds more like an individual while 日本の女性 more collective. (Somewhat similar to a Japanese woman vs. Japanese women, but maybe not always.)
For example, 私は日本の女性が好きだ means I like Japanese women in general; 私は日本人女性が好きだ is acceptable, but it would be more natural to use it when you like a ...
空いて【すいて】いる means the restaurant is not crowded (there are many empty tables), whereas 空【から】だ means the restaurant is completely empty (there are no customers). The same is true also when the subject is a theater, a train or a theme park.
Note that the correct reading of the latter is から. そらだ ("is the sky") is nonsense in this context.
You may also ...
乗る on its own only describes the action of mounting or boarding itself. The "destination" marked by に must be a vehicle (or a boat, a horse, etc), not some geographical location. 東京に乗る or 仕事に乗る does not make sense (although "to ride to Tokyo" is a valid expression in English). 車に乗る does not necessary mean you travel to somewhere; you may ...
Let me explain 歌謡曲 first. Today, 歌謡曲 is a word that refers to "Japanese pop songs roughly in 1930-1970". 昭和歌謡 refers to almost the same thing. Wikipedia defines this like this:
Kayōkyoku (歌謡曲, literally "Pop Tune") is a Japanese pop music genre, which became a base of modern J-pop. ...
Your translation attempts will never mean "to like to show" or "to want to show". Something like this is expressed using a completely different construction in Japanese. Do you know about the -tai form?
However, your attempts have several other mistakes.
Adjectives like いい don't take だ, so いいだと思う is ungrammatical.
好きになる means something ...
「おえる」is paired with「おわる」。Somewhat contrived example:
Highschool has ended.
I have finished high school.
These two sentences are basically the same. Even if you use the active voice "finish", you don't have any control over when your highschool education ends. (Okay, you do have some control but not much.)
So, even though おえる is ...