To your first part, the sun does influence climate, but the science has found temperatures still increase, even when solar cycles are at their lowest point. If global temperatures strictly followed solar cycles then temperatures would periodically go down to match them.
With respect to CO2 lagging:
To claim that the CO2 lag disproves the warming effect of CO2 displays a lack of understanding of the processes that drive Milankovitch cycles. A review of the peer reviewed research into past periods of deglaciation tells us several things:
Deglaciation is not initiated by CO2 but by orbital cycles
CO2 amplifies the warming which cannot be explained by orbital cycles alone
CO2 spreads warming throughout the planet
CO2 lags temperature - what does it mean?
That was a whole bunch of rhetoric just to say 'we hate executive summaries'.