I think so, there is a lot of correlation between few sunspots and cold temperatures although the mechanism of how this occurs have not been discovered.
The "total solar irradiation" TSI changes little between solar maximum and minimum, which has led some scientists to dismiss this, however there are other changes, extra infrared changes quite a bit, as well as solar wind, and AP magnetic index, how these could effect climate has not been determined as yet, but the coming minimum and next cycle "25" should help us find out.
The science is inconclusive regarding the affect of sunspots on atmospheric weather. Variation in solar radiation has a huge impact on space weather which impacts our radio communications systems. An interesting anecdote: the "Little Ice Age" occurred over parts of Earth during the Maunder Minimum (1645 - 1715) a period of almost no sunspots.
Sunspots are correlated with solar output, and when solar output changes it changes the weather and climate. I don't believe it's the sunspots per se that affect things.
Not directly. Sunspots are a proxy for TSI which does effect weather and climate and for cosmic ray flux, which may effect weather and climate. You need to remember that cosmic rays are not the only source of seeds for clouds. Dust probably acts as a negative feedback for the effect of cosmic rays on clouds, but it probably has a lag time of several years. A lack of cosmic rays could have caused the dust bowl.
Yes. Sunspots reduce cosmic ray intensity on earth. Evidence is that cosmic rays seed clouds. Thus, higher solar activity reduces cloudiness which increase surface temperatures.
The effect is temporary, as solar activity has a cycle of about 11 years. It has only a fraction of the effect on temperatures as does changes in the greenhouse effect. It has created year-to-year differences but it does not explain the ongoing trends of increasing heat in the troposphere and decreasing heat in the stratosphere.
