Regardless if the adapter is Dual Link or not doesn't matter. HDMI is electronically identical to a Single Link DVI so once you involve an HDMI cable in the connection, a single link is all you get. HDMI cables simply do not have the same amount of conductors as Dual Link DVI. Now, that doesn't mean you won't get a video signal to the monitor with an adapter and an HDMI cable. The question here is how the monitor utilizes the second link. If the monitor's signalling supports HDMI 1.4b over the primary link and uses the second link to go beyond, then you'll get what HDMI 1.4b can support.
Looking at a side-by-side of a Single Link DVI and HDMI connector, they have the same amount of conductors. Also, it's not quite true that you can't carry audio if any adapter is involved. While it's true that around the Radeon 3000 and Geforce 9000 days special adapters from the manufacture were needed to multiplex SPDIF audio and DVI video together, no longer was the case once video card manufacturers included an audio codec into their designs. I have a very old DVI to HDMI cable running from my PC to my audio receiver. Works just dandy. There is no specific 'Audio' pin with HDMI, all just one data stream.