Does the Sun's oscillation affect Earth's climate?
I'll get you started at least, this is a bit too broad otherwise; I guess what you're describing are called Milanković (Milankovitch) cycles. Are climatologists aware of these? Yes, for a long time. Milanković published his first paper on the subject, Theorie Mathematique des Phenomenes Thermiques produits par la Radiation Solaire, in 1920. As for Solar oscillations in general, even much longer periodicity of ~ 26 million years of the Sun's oscillation about the galactic plane has been considered before (in 1984). As we're aware of the 11 year solar cycles, and have been integrated into climate models that climatologists work with, including collective effects of all of these manifesting in grand minima and maxima of solar activity (PDF).