Let's tackle these one by one:
It's very important for me/to me to study abroad.
Here both can be used, but they would have slightly different meanings:
- "To me" means that you're the one who cares about it, while "for me" doesn't necessarily mean that it's important to you - it could be that it's important because of somebody else.
To better understand the difference, let's add "My dad thinks":
"My dad thinks it's important for me to study abroad."
...means he thinks it'll be a good thing for me.
"My dad thinks it's important to me to study abroad."
...means he thinks I care about studying abroad.
For me/to me all that matters is how much it will cost.
- First off, I would say the word order here could be improved: I'd recommend going with "All that matters to me/for me is how much it will cost". The order you used puts a stress on the first part: I'd expect to see these sentences in comments such as "I know you care about the time it'll take, but for me/to me all that matters is how much it will cost."
Secondly, technically the same rule as above also applies here: "to me" means directly to you, "for me" might mean that others are the ones who care, though in this case this difference is very small, and somehow the "for me" sounds a tiny bit worse or more awkward. So:
All that matters to me is how much it will cost.
= eg. I don't care about the time it'll take, I just care about the price.
You are the most beautiful girl, at least for me/to me.
- In this case, only "to me" sounds good. Perhaps somewhat because of the distinction noted above - we're talking about the speaker's opinion.
In the second version, let's add on a beginning:
"My mother thinks you are the most beautiful girl for me."
Here - and this is a reach - but it could technically be understood to mean that my mother thinks this girl is the prettiest I can find, though there are prettier ones out there in general
I imagine some might use such a sentence as a concatenation of two ideas: "you are the most beautiful girl" and "you are the girl for me", but technically this would not be correct.
On the other hand, in this sentence:
You are the perfect girlfriend, at least for me/to me.
...the better option would probably be the first one, but again, there would be a slight change in meaning:
"...for me" would mean that I'm acknowledging that while others may also have (near)perfect girlfriends, my specific needs, character, interests, etc. mean that you are the one who is perfect for me. However, note that the girlfriend would prefer to hear this as "You are the perfect girlfriend for me"
"...to me" could be understood to mean something along the lines of, You weren't (or aren't...) the best girlfriend for your other boyfriends, but I think you're perfect. That said, in this case you can't skip the "at least" - it wouldn't sound good without those two words.
To summarize, no wonder you're having problems, because there's a lot of nuances! But I hope this will have helped a bit :)
In addition to the issues regarding proper nouns mentioned in this question and its answers, there are some usage notes specific to the words you've asked about.
Home
I'm going home.
The speaker is going to their home.
I'm going to a home.
"A home" would be usually be interpreted as some sort of assisted living facility. A person with mental illness or an elderly individual may be put "in a home".
I'm going to the home.
I would not know what to make of this sentence. The home? Which home? What category of homes are we talking about that I would automatically know to which one you're referring? You mean your home? The home you put your grandfather in? What?
School
I'm going to school.
If the speaker appears to be leaving and heading elsewhere, then I would conclude that they are on their way to the school that they attend.
If the speaker is staying put, then I would take this to mean that they are enrolled in classes at a school.
I'm going to a school.
I would probably take this to mean that the speaker is enrolled in classes, but would find the phrasing a bit odd. I would likely ask them which school.
I'm going to the school.
Similar to 'home'. Which school? What are you talking about?
Perhaps this would make sense if there was a particular school that was obvious from context. You might also say this if you were going to a school building for a reason other than going to classes, in order to emphasize that distinction.
Airport
I'm going to airport.
This is definitely wrong, unless the speaker was actually referring to some place named 'Airport', like a bar or club.
My first instinct would be to ask if that was a street name.
I'm going to an airport.
This would be an odd thing to say, but would be understandable. It's odd, because one rarely wants to go to just any airport. It's not like a supermarket where you can find pretty the same stuff whichever one you go to. You need to go to the airport which has the flight for which you have tickets.
Maybe someone might say this if going to any airport fulfilled some interesting criteria, like maybe the first time they had been to any airport, or some fortune teller told them they'd meet their future spouse in an airport.
I'm going to the airport.
This is normally what people say. This is because there is usually only one nearby airport, which almost everyone in the region uses for all of their air travel.
If there were multiple airports nearby that one might potentially use, then you would use the airport's actual name instead.
Best Answer
You could use "at a discounted price", "at a discount", "with a discount", "on sale", "at sale pricing" and "on clearance."
Discount is a general term, but is also used for special deals that only you can get, like educational discounts for being a student, or a discount from the store for shopping there for so long. Sale is for discounts anyone could walk in and get. (Sale can also just mean that it is available to be bought, so be careful for confusion.) Clearance is for goods that the shopkeeper doesn't want in stock anymore, and so the shopkeeper gives them a big discount so they sell quickly until they're all gone, even if it means losing money compared to the price they paid.
I don't really like your second sentence, "with a discounted price" doesn't quite come together. It's just a little off. You're also right about "in", I can't think of a single phrase that would work with.