Airport Express - WiFi network extension with wireless speakers
Yes you can use the Airport Express to extend the wifi signal from your other router:
You just have to change the settings for your device in the Aiport utility to extend a network instead of creating a new one.

http://support.apple.com/kb/HT4145#

Yes you can use your speakers wirelessly. You just need to connect them to Airport Express. Apples technique is called Airplay:
https://www.apple.com/airportexpress/