I agree with pegminer; this is an excellent question, deserving of the stars.
There are some excellent answers so far, particularly from linlyons, pegminer Trevor and AMP. I cant really add to those, so I will try a different tack: A simple analogy!
It's not perfect; it does have limitations, but it is one that will be familiar to many so, hopefully, more easily understood; the graphs of y = cos x and y = sin x
http://educationalstationery.com/images/scos.gif
In a nutshell, the red curve (y = cos x) represents the path of the sun and the positive value of y its relative strength. The x axis is the horizon.
The blue curve (y = sin x) is the temperature response.
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I'll spell it out, in case anyone isn't quite sure ...
The y axis (x = 0) corresponds to around noon. The sun is at its highest and strongest and the temperature, though below its peak, is rising at its fastest. (the temperature curve is at its steepest)
As we progress through the afternoon, the sun gets lower and its relative strength (the forcing) gets less. Its still a forcing though, and the temperature continues to rise, but the rate is slowing. (the curve is leveling off)
As we approach 6pm (90degrees) the sun nears the horizon, the forcing nears zero and the temperature curve flattens out. Peak warming has been reached.
As the sun slips below the horizon, the forcing disappears and rapid cooling sets in, continuing through the night.
At 6am (270 degrees) the sun rises, the cooling levels out and temperatures start to rise again, but only slowly at first as the strength of the sun is still relatively low.
As the morning progresses, the sun gets higher, the strength of the forcing increases and the temperature rises more rapidly in response.
At noon the cycle starts again.
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As I said, this model could be improved, but it illustrates the main points, and I think it gains something for its simplicity. Hopefully it explains the first part of your question.
Regarding the question of the lag after the solstice, it's basically the same idea as above.
The difference is that there is a very large change in forcing between day and night, and a very rapid initial temperature response. When looking at (say) a year, the days and nights aren't really relevant; we just average them out. The change in forcing between summer and winter is less than that for days/nights and the rate of change is much slower, as is the response.
It's the same again for the solar cycle. It's not correct to say that there is no time lag for the solar cycle, it's just that the change in forcing over a cycle is so slight that the response would be negligible.
It would be as a ripple on the shoreline; swamped by the waves of day/night, the tide of summer/winter and the Tsunami of El Nino!
Don't get me wrong, I'm not saying a succession of weaker or stronger solar cycles wouldn't have some effect on temperature, but the change over a single cycle, I'd be surprised if it was even detectable.
Please see my answer in the question referenced by AMP for further details.
https://answersrip.com/question/index?qid=20110615120328AAQh1XY