1. 

Yes, it is a leading theory of climate change. http://www.sciencebits.com/CO2orSolarHYP...

The relationship is complex. The total solar index (TSI or PMOD) has only been measured since 1979, and has not correlated well with global temperatures. However, an older index, the Sun Spot Index (SIDC) that is a reasonable proxy for PMOD goes back centuries, and correlates much better. http://www.woodfortrees.org/plot/hadcrut...

Note that the SIDC correlates with PMOD, but not the global temperature since 1979, but this short 34 year period of non correlation is not unusual. Similar periods exist between 1880 to 1900, and 1940 to 1980. Going back to the Little Ice Age, we find that the drop in global temperatures corresponds imperfectly with a low in the SIDC known as the Maunder Minimum. http://en.wikipedia.org/wiki/File:Sunspo...

It is interesting to compare the SIDC with temperature proxies. http://www.econ.ohio-state.edu/jhm/AGW/L...

Then, the older temperatures can be compared with a proxy that goes back further, the C-14 activity. http://commons.wikimedia.org/wiki/File:C...

Taking a close look at those graphs demonstrates a viable driver of global temperatures that is far from perfect. However, the other theories do no better. http://www.john-daly.com/

There are things to consider:

1. It is not clear which solar output is really responsible for the correlations and how they interact. Cosmic rays appear to be at least as important as the total solar output. http://repository.ias.ac.in/42732/

2. Changes in solar output caused by the sun heating up or cooling off alters the wavelength of the peak output of the sun, [2] changing the relative effects of greenhouse gases. http://en.wikipedia.org/wiki/File:Atmosp...

3. Be careful about accepting unverified outputs from computer models based on solar drivers. [3] These have proven just as unreliable as the IPCC's CO2 driver models. [3a] Since they did sucker many CO2 skeptics into believing them, they also proved that Skeptics are just as capable of idiotic unskeptical religious thinking as the Warmists demonstrate.

4. Few people, especially not climatologists understand the full complexities of the solar theory. There are time lags between output and effect that vary depending on ocean currents. There is also the problem of interaction between greenhouse and anti-greenhouse gases whose effect depends on concentrations and altitude. [4]

The Sun is the power source. Other things being equal, changes in the energy output do result in changes in Earth's temperature. The Sun's magnetic feild also deflects cosmic rays, which may seed clouds.

Greenhouse gases and the urban heat island effect also effect temperature by making Earth more effective at absorbing the Sun's energy. Dark surfaces, like parking lots absorb the energy from the Sun and greenhouse gases absorb energy from Earth which would otherwise escape into space. The reason why Earth is warming while the Sun is cooling is that changes in carbon dioxide is more dramatic than changes in solar output.

Here's a couple vids. you may find interesting. (May have to rewind).