The optimum depth depends on physical/budget constraints as well as the owner’s goals for the pond. A pond for swimming would be managed quite differently than a pond managed for trophy bass fishing. Many, if not most, small ponds are affected by nuisance algae or aquatic plant growth. It is important to identify the type(s) of nuisance algae/vegetation as the control methods will differ. As you and Philospher have hinted, deepening the pond may help.
As the pond gets deeper, you start to have a zone of deeper water where emergent (rooted) plants and filamentous green mat-forming algae cannot get enough light to grow. Depending on the size of the pond, local climate, stream/source/fountain influences, and how exposed it is to wind, it may also begin to experience thermal stratification - separation into layers based on temperature. This may help because much of the nutrient recycling occurs at the bottom where decomposition is taking place, while sunlight is only available nearer the surface. A stratified pond also provides a low light refuge for zooplankton- microscopic animals that eat suspended algae. Zooplankton are fed upon by sight predators such as sunfish, so they may retreat to the darker water during the daytime and feed at night, keeping suspended algae in check. Encouraging a population of predator fish to keep sunfish numbers in check is a “top-down” approach to controlling suspended algae that has the added benefit of increased recreational fishing opportunities.
It is often not feasible to dredge or otherwise deepen a small pond, so there are other options that a pondowner facing this situation might try. The primary concern should be excess nutrients that are fueling the algae growth. Instead of mowing grass right to the edge, the pond margin should be left alone, or better yet planted with native vegetation. This keeps nutrient-laden runoff from getting into the pond. When abundant, Canada geese can be a significant source of nutrients to a pond, though ducks and swans are usually not a problem. If neighbors are feeding geese, they should be educated about the problems that can be caused by overabundant geese. Once geese become established, they can be difficult to manage.
There is a non-toxic treatment called Aquashade that dyes pond water blue. The dye intercepts light and can help keep algae down. Another non-traditional treatment is barley straw- bundles of decomposing barley straw have been shown to have a positive effect on algae ridden ponds and many pond dealers sell straw or barley extract for this purpose. You may also see bacterial formulations advertised- they are purported to contain bacteria that can help clean up a pond very quickly. Beware of any product that seems too good to be true and keep in mind that bacteria are basically ubiquitous in the environment and there is usually no need to “seed” them.
Algaecides have come a long way from the use of plain copper sulfate and there are many choices available, from spot treatments for pond weeds to “whole pond” chemicals that can gradually and safely reduce suspended algae. It is important to identify the target vegetation and follow all labeling instructions when using these chemicals, especially if the pond has an outlet to a stream or other waterbody.
You mentioned mosquito breeding, but mosquitos are not often found in ponds for the very reasons you and other posters have mentioned. They are far more likely to breed in ephemeral sources of water such as tires, cans, pails, clogged gutters, etc. Even without fish, there are usually plenty of beetles and dragonfly predators to prevent much mosquito activity.
I am an aquatic biologist and ponds are one of my specialties. If you post some more details about specific problems with the pond I may be able to help you further or at least steer you in the right direction.