In another thread I asked about the meaning of dışarısı in a sentence, and why it had that form instead of dışarıda. I looked them up it the dictionary, and they mean more or less or even exactly the same.
Are they really interchangeable?
I think it was Meltem who said that you have to look at this word as if it was a noun, and then add state suffixes just the way you would with a noun. I thought that was a very clear explanation. But I don't know when to use -ı and when to use -da..

dışarı
1. out.
2. the outside, exterior.
3. the space outside.

dışarıda
1. outside, out of doors, outdoors.
2. abroad.

dışarısı
outdoors, outside.


Thanks
Elisa

Dışarısı is indeed a noun and it means "the outside".

Dışarısı karanlık.
The outside is dark.*

Dışarısı soğuk.
The outside is cold.*

*I know the translations sound unusual but I had to stress "the outside".

dışarıda is the locative case of the noun dışarı. As all locative cases this one is the answer of the question "nerede?" or "kimde?". Nerede? means in what place/location/direction?

Ahmet nerede?
(Ahmet) dışarıda.

Dışarıda is the answer or the question in what location/place/direction

I think I saw the light: dışarısı is a noun that is based on dışarı, which is a preposition?

Dışarı, dışarısı or any other version of dış can take some case suffixes because these are noun based words (isim soylu kelime). When they modify a verb they are called adverbs.

Dışarısı is strictly a noun and is never used as an adverb. Dışarı is both an adverb and a noun. All versions of dışarı and dışarısı that have case suffixes are nouns only. When a case suffix is attached to a noun it cant be used as an adverb.


Examples:

Ahmet dışarı çıktı. > Adverb
Elimdeki kitabı pencereden dışarı attım. > Adverb

Dışarısı çok güzel. > Noun
Dışarıda güneş parlıyor. > Noun
Dışarıya çıkalım mı? > Noun