Does precession mean that season start points will switch in 13,000 years?
Yes, the positions in the Earth’s orbit corresponding to particular seasons will change over the precession period- see answer to above question (and also the next one). If the axis of rotation will be opposite on the precession circle in 13,000 years, does that mean that, relative to our current calendar, the seasons will be opposite, that summer in the Northern Hemisphere will be in December, while winter will be in June? Well, positions in the Earth’s orbit corresponding to particular seasons will indeed change over the precession period- see answer to above questions. However, in fact our calendar is trickily defined to keep the seasons stable with respect to months of the year, as we’ll see later.