Airport Extreme to 3 Sony Airplay speakers

what you want to do is setup your express to be a "bridge" for your airport extreme network, you want to create a "new" wireless network (but if you use the same settings as your extreme, most macs should seamlessly work and switch as required)

make sure your airport express is configured not to do any NAT or give out IP's as that is maybe what is conflicting with your network currently, also make sure you give it an IP in the same subnet as your airport extreme and also that it does not conflict with any other machines

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/