Learn English – Which expression is correct “in purpose” or “on purpose”

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.

I can't think of any English term that is regularly used for this – at least, not in a noun form.

As Steve Ives mentioned in his answer, this is how I would ask the question:

What time are you leaving?

The term departure time is grammatically correct, but that expression is normally reserved for transportation: planes, trains, and busses. And it generally refers to the time that the conveyance departs (everyone knows that passengers must arrive earlier than that so they can all board before departure.)

If you want to tell your friends that you'll be leaving the campus at 5 o'clock, you'd say something like:

I should be leaving the campus around five.

or:

I should be done at the campus around five.

You'd only use departure time if you were trying to be funny by making something ordinary (i.e., leaving the campus) sound very official:

My departure time from campus will be 5 o'clock.

I could say a father saying this to his young children, if they were leaving for a trip the next day:

Our departure time will be 8 AM.

but that would be overly formal and therefore deliberately humorous. A more natural way to say it would be:

I want us to get out of here by eight.

or:

Let's be pulling out of the driveway before 8 AM.

The same holds for arrival time. We typically say something like:

We should be there a little before 10.

not:

We have a 9:55 estimated arrival time.

unless we are trying to be somewhat humorous by being overly technical.