What is the importance of mixing caused by outgassing in the ocean?
Outgassing is a process by which $O_{2}$, $CO_{2}$ and other chemicals are released as a result of heat or increase in temperature. It is different from advection and diffusion which are mixing processes as a result of temperature and concentration gradient. Outgassing is an abrupt release of gases from where they are stored trapped dissolved, in this case  $O_{2}$ and  $CO_{2}$ in the ocean. Like the ocean some materials are prone to outgassing when heated. For example chemicals released from plastics (e.g. the odor of a new car). 


  What is the relative importance on mixing of the vertical flow of
  gases in the ocean water column?
  How does the mixing compare with physical mixing (e.g., wind, internal wave breaking) and molecular diffusion?


The rate of outgassing in ocean is comparable to molecular diffusion of the gases. However, the mechanisms of outgassing and diffusion are different,i.e, outgassing occurs due to some sort of pumping mechanism but diffusion occurs due concentration gradient. Both mechanisms are important to stratify saturation,i.e. homogeneous distribution of chemicals. 
Having said this, comparison of diffusion and outgassing needs further investigation. However, I don't think outgassing like diffusion can be triggered by saturation. Some kind of pumping mechanism attributed to difference in surface layers (e.g. the creation of temporary vacuum layer) may trigger outgassing. 
In connection with climate change, ocean oxygen and carbon dioxide outgassing into the atmosphere are decreasing ocean oxygen and increasing atmospheric carbon dioxde. Both events have negative consequences. Decrease in ocean oxygen has impacted ocean plants and animals. So, outgassing if occurs within the ocean can be taken as a mixing process but if it throws out the molecules out of the ocean has negative consequences both for the ocean and atmosphere.