Mendeleev's predictions
Well, guess what, even I study the exact same textbook (class 11 NCERT, from India). I had asked my teacher the exact same doubt, and the answer was:

Due to its proximity with neighbouring substances, like Arsenic (which was already discovered by then - see here), Mendeleev was able to accurately calculate the average values of density and melting point of Germanium, or eka-Silicon.
The formula of its Oxide and Chloride was predicted with respect to the formula of the Oxides and Chlorides formed by other elements of the group IVA (as per old IUPAC numbering), such as $\ce{CO2}$, $\ce{SiO2}$, $\ce{CCl4}$ and $\ce{SiCl4}$. Their densities were calculated using the knowledge of Chlorine's and Oxygen's atomic weight, and the predicted density and/or atomic mass of the new element.

Although I don't know how the boiling point of a substance can be predicted, my best guess for it is that it was approximated using the boiling point of other similar compounds near it in the Periodic table. For more information on how Boiling point can be possibly approximated, here is a link.