This is an interesting question & I've been waiting to see what answers, if any, would be written.
One affect of creating such a lake would be a localized increase in humidity in the vicinity of the new lake, but nothing that would significantly increase rainfall.
By bringing in salty sea water there could be a risk of contaminating existing underground fresh water stores with salt, making such sources of water unusable.
Also, by being salty, the water could not be used irrigate crops. It may be possible to establish a fishery in the lake, bringing in salt water species of fish, but that would need to be carefully managed.
If for whatever reason, the supply of water was not maintained in the lake, eventually the lake would dry up leaving salt behind. This would then be blow around by the wind, contaminating the nearby land with salt and killing plants on that land and making the land unsuitable for any crops that may grow there now.
Also, if dew were to form, there is no control over how much dew would be created. Many factors come into play when dew forms: atmospheric pressure, wet and dry bulb temperatures, the amount of moisture in the air (humidity) and enthalpy of water vapour and liquid water.
Edit
Having thought about this some more, something similar to what you are proposing has already been done in Egypt, when the Suez Canal was completed and opened in 1869.
Within the canal system is Great Bitter Lake, which is a sea water lake. The presence of such a body of water for nearly 150 years has not increased the amount of moisture in the vicinity of the lake that would allow lush vegetation to grow.
I'm not sure that salt at the surface would be a significant problem. Much of Australia has ephemeral saline lakes at the surface that don't have major impacts (in my opinion) on the surrounding land when they are dry. But you are talking very large scale. I expect that the dry lake bed would stabilise with a sediment cover blowing in so salt dispersion would cease.
1) Would it work? I think it's a great idea but I'm skeptical. There are a lot of areas near the oceans where dew formation doesn't seem to produce significant amount of green vegetation. Well there are parts of Chile where dew is important in supporting the ecosystem, but that ecosystem is pretty unique and sparse. Not something that would support much human population.
If I understand your question you are suggesting that the green vegetation would produce a feedback loop that promotes further dew or precipitation and becomes self sustaining. I can't say whether that could occur in some regions. If you look at the Great Artesian Basin of Australia, Lake Eyre and other very large lakes will fill with fresh water periodically and then go saline or even dry out. However, they haven't developed self-supporting vegetation, to my knowledge. The terrestrial plants rely on periodic rainfall due to storm tracks, generally from the west or north. If nothing else, thinking about it will teach you a lot about climate and weather. Hope this helps.
But the key underlying question here is: What for? The fact that something is big, cheap, and doable does not make it worthful. The environmental implications of these huge geoengineering projects are enormous and difficult to predict in detail.