The first one is rather simple: You apply to a company/organisation etc. for a post/a grant etc. In other words, you ask someone (eg. a company) to give you something (eg. a job). So you'd say:
-I applied for the job.
-I've already applied twice to them but I never got an answer.
You can be angry with/at someone:
-I'm very angry with you.
-She was angry with/at the salesman.
You are usually angry at things/conditions etc.
-I'm angry at the weather.
Angry with usually implies that you can interact with someone, ie show them that you're angry, shout at them etc. It also means that the other part can know or feel that you're angry with them.
Angry at only shows the object of your anger. So, you can't be angry with the weather, because it is not a person and you can't make the weather feel your anger (unless of course you wish to personalise the weather, which many people do...)
Look back at the first example (I'm angry with you). With is preferable to at here, because you're in direct interaction with the other person and he/she can feel that you're angry. If, on the other hand, you say She was angry at the salesman, it's probable that she didn't directly or expressly show her anger.
Anyway, this is hardly a rule and people often use either at or with (when a person, or even an intelligent animal - such as a dog - are involved). But you'd usually prefer angry at when speaking of inanimate (not living) things.
Note also that you can say angry with someone about something: She was angry with me about the money.
You arrive at small places, such as a bus stop, a railway station, an airport, the city center etc., but you arrive in large places, such as a town, a city, a country, a continent. The border between the two choices should be a place more or less the size of a small village. For a small village you can usually say both: I arrived at/in the village.
You are usually bored with (also: bored of, especially - but not necessarily - when a gerund follows) things or situations which are boring: bored with/of going out with the same people all the time, bored with the political situation.
You are often bored by someone or something, especially if they impose something boring on you, something you can't easily choose to avoid: I was bored by the chairman's long speech.
Usually, however, there is little or no need to show how exactly your boredom was brought on, so all of these prepositions can more or less be used without any change in the meaning.
Both to and with are often used with the meaning remaining the same. If we need to find a difference, though, the following could somehow describe it:
-Remember Sinead O'Connor's song Nothing compares to you? Well, that should mean that nothing can be even close to what you are, you're much better, much more desirable, much more of what I'd love to have.
He compared the beautiful girl to an angel (ie to something of a different quality).
-Compare sth with sth else, on the other hand, just means that you put things side by side in order to see their differences or similarities: You can't compare this cheap ring with the pearl necklace the old man gave her (ie you cant' say they're anywhere near to being similar). The police compared his fingerprint with hers in order to find which of them had committed the murder (when two things are hard to tell apart).
The distinction is far from easy to make, but it's rarely important either. Very often you can use both for the same situation. Experience will teach you when to use one or the other in those few cases when it makes some difference.
When you care about someone it means that you want them to be well, happy etc. and you do your best for their happiness, safety, well-being etc.: Your parents are doing this because they care about you.
If you don't care about someone/sth (sth = something), you are indifferent to them, it makes no difference to you what happens to them.
To care for means:
-To take care of someone: She couldn't go on holiday because she had to care for her babies.
-To like someone, especially in an affectionate way: You know I care for you, Jane.
-(Used to speak in a polite manner) want: Would you care for a cup of coffee?
Cluttered with = filled with things in a disorderly manner: The room was cluttered with books, toys, clothes etc.
Student or Learner