How can a Mac mini drive 3440 pixels wide display instead of 1920?
Yes. The limitation is in the adapter cable. The one you listed should work well and there are many others now like the cable matters one linked below. 


http://www.apple.com/shop/product/HHLY2ZM/B/belkin-4k-mini-displayport-to-hdmi-cable-2-m-65-ft?fnode=8b
http://www.amazon.com/Cable-Matters-DisplayPort-Thunderbolt-Compatible/dp/B00K0UDJFI