These 3 sentences all translate 冲着 as 'at'. You could do that too.
Was that remark directed at me (冲着我)?
Then he started shouting and swearing at her (冲着她).
He had her by the arm and was screaming at her (冲着她).
Rat shouted at Mr Timid (冲着胆小先生).
(Rat you rat! Leave him alone!)
The character 「衝」 was used early on for a meaning like 「衝擊」 (to charge at someone/something, to assault someone/something). By extension, 「衝着ＸＹ」 simply means direct [a type of emotion] towards [X], [Y]-ing where
For the majority of the time, a type of emotion is usually aggression
X is the object towards directing this emotion
Y is the actual manifestation ...
冲着 in this case means towards. 'Toward' itself can be regarded as an action, even though it may not be in the literal sense of 'movement'.
冲 indeed generally means physically rush. However, in case of 冲着, it is never physical, but directional. Compared to 对着, 冲着 implies a sense of urgency and harshness.
A simple example:
我对他喊 vs 我冲他喊
Both means 'I shout to ...
There is no rule for this and is mostly based on preferences. If the verb for the two actions are the same, you can choose if you want to omit it. However, in China, often one way of expressing is more fluent and more widely used than the other. The exact rule is difficult to describe, but it is likely based on the rhythm and flow of the language.
From C. Li, S. Thompson, Mandarin Chinese: A functional reference grammar:
This category of questions explicitly presents the respondent with a choice of two or more possible answers connected by 还是, which may be called constituents. The syntactic nature of the connected constituents may vary from question to question, but all such constituents within a ...
体贴 (to be considerate) is intransitive, it cannot take a direct object. So you can be "considerate towards your parents" but you cannot "considerate your parents". In a similar vein, you can be "happy for someone" but you cannot "happy someone".
关心 (to be concerned about), on the other hand, is transitive. It can take ...
I saw this question from maths stackexchange. Although I'm not an expert and just come by, I want to say something: This 的 can be there but it's NOT necessary(from the perspective of a native Chinese speaker, and I'm not sure if in your Chinese test this would be acceptable). The flexibility of Chinese might not be very comfortable for English speakers(which ...
的 + noun;
得 + adjective;
地 + verb
the second 的 in this sentence should be 得. Sometimes we know it should be 得 but cannot be bothered to correct it in the informal comms as they have same pronouciation and we understand its meaning or the context behind it.
车外： Object + related location/position = place term, n.
的： Aux. term to turn n. into descriptive, usually in possession/belonging relation; or habitually added after multi-character adj. (single character adj. usually is optional with 的)
阳光： n. subject of the sentence
格外： itself is adv. no need to add any aux term. 月亮格外明亮/汉语格外难学
的： Double mistake - no need ...
车外 (outside of the car)
的 (adjective marker)
"车外的" (of the outside of the car) is an adjectival phrase that applies to the noun '阳光' ( sunshine)
刺眼(的) is the adjective for '阳光' with 的 omitted
格外 (especially) - 格外 is an adverb, the 的 in 格外的 is for emphasis and can be omitted
的: 助词, 同 “地”（de）。用在状语后，表示状语和中心词之间的修饰关系)