in selecting subwoofer for music, it depends on what kind of music do you usually play.
if you want audiophile sweet sounding music, choose sealed sub (sound: tight and punchy) like wharfedale subs priced at 12k at listening room which is within your 10-15 budget.
if you want rock or other loud music, choose ported sub which can produce loud and deep bass.there is a polk audio sub within your price range.also within your range is pilyo sub with amp4v2 by marco pilyo priced at almost 12k.you can also try the sub of anthony audio which has the 10" or 12" woofer.
using speaker input or high level input results in double filter of the sound, first by the AVR then by the sub. hence, it should be cleaner. but the sound difference is only slight.