Listen Again seems to use RealPlayer. It will open the iplayer window, and you can right click on the bottom section and should get a menu saying Play in RealPlayer. Listen Live seems to use MediaPlayer. Stream seems to be www.bbc.co.uk/radio4/wm_asx/aod/radio4.asx so you could try opening that directly in MediaPlayer.
Yes, listening to radio will use some of your broadband limit. Probably best to monitor how much you use. If your broadband provider does not let you see amount of data you have used, try installing NetMeter which will monitor your downloads and uploads and warn you if you get near a specified limit.