Hi Martin,
Your ir is modulated with aprox 38 khz, so using the sound card is the wrong way to start.
From your picture I don't understand how you captured it, but lets start the way I did it:
My advice is to spend 2€ on a 38khz ir receiver with digital output and a ir led and install the library from the second post.
You should be able to see the byte stream and play it back. If your air conditioner responds to it you have a solution using certain preconfigured commands.
Decoding is another story:
The library will give you the streams in a numeric format that can be converted into a 1/0 stream (with ms excel)
You set the temp to 29 and then to 28 and look for the differences in the stream...
Let me know when you have the hardware tested with the library and give me the output from ir library, I'll fill in the excel and post it.
One word of caution: the remote is the state machine memory, if you use another remote in parallel your arduino is quite lost.
The code I have until now are some undocumented test, but the main is irsend.sendRaw(codesArray,buffer_size,38);
Regards,
Adi