Ad vs. Af
You can't hear the difference between ad and af when pronounced, but most of the time you will use af. There are a lot of danes who get this wrong too, so don't panic if you don't know which one to use ;)
Basically you need AD when we are talking about a movement along with something or through something.
- De gik hen ad vejen (They walked along the road)
- Væsken løber ned ad muren (The liquid is running down the wall)
- Han gik ud ad døren (He walked out the door)
- Vi kælkede ned ad bakken (We were sledding down the hill)
- Fuglen kigger ud ad vinduet (The bird looks out through the window)
In all the examples we can replace the "ad" with either langs (along), igennem (through) or via (via).
On other occasions you need AF. Typically when we are talking about a movement from or away from someting, or a cause of something. Most of the times you can replace "af" with the word "fra" (from/of) and it will still make sense.
- De græd af sorg (They cried of sorrow)
- Huset var bygget af træ (The house was built of wood)
- Fodboldspillerne kom ud af bussen (The football players came out of the bus)
- Jeg gik ud af huset (I walked out of the house)
- Hun smiler af glæde (She is smiling of joy)
- Flasken var fuld af vand (The bottle was full of water)
- Tag låget af gryden (Take the lid off the pot)
Another example would be the following:
- Toget kører ad sporet Means that the train is running along the track
- Toget kører af sporet Means that the train gets derailed and is heading towards an accident
Some external sources in danish:
- Video: Af eller Ad http://www.bedredansk.dk/rettenoeglen/forholdsordsfejl/af-eller-ad
- Worksheet with solutions http://www.grafisk-litteratur.dk/dansksprog/Ad%20eller%20af.pdf
This is a very good question. And the simple answer is that "kigge ud af vinduet" is wrong.
The thing is that no one pronounces "ad" as it is written. Most of us pronounce it as "af" (or basically just "a") and a lot of people do not make the distinction between "ad" and "af", myself included, as I had to look up the difference between the two.
The right uses of these are:
"Af" is used when it can be replaced with "fra" (from).
"Ad" is used when the meaning is "along, over or through something"
There are some exceptions though. Fx. If you are laughing at or smiling to someone, you can use both "ad" and "af"
You can read more here (it's in Danish though)
I live in America, and I think that it is definitely an acceptable sentence. I think that you are confusing the meaning; Looking THROUGH the window is not quite the same as looking OUT through the window. Looking out through the window implies that you are inside and you are looking outside.