Do apps stop running when I close the lid?

Yes everything stops when the lid is closed and the MBA goes to sleep. There will be no further activity with Skype so if Skype detects the inactivity it will probably terminate the connection. The only thing happening during sleep is the hardware monitors the battery. If it sees the battery get too low it will come out of sleep and shut the MBA down in deep sleep (hibernate) mode.

PowerNap allows the computer to perform some actions even if it is put to sleep but this doesn't extend to applications. So your computer can check for mail and perform other authorized tasks but an application like Skype cannot function until the computer wakes up.