Can hydrazine be protonated twice?
Yes, it is possible, but only to a very small amount.

The $\mathrm pK_\mathrm a$ of the hydrazinium (2+) ion in water is $-1$ (CRC Handbook of Chemistry and Physics, Ed. D.R. Lide, CRC Press, Boca Raton, FL, 2005), so the equilibrium

$$\ce{N2H5+ + H2O &lt;&lt;=&gt; N2H6^2+ + OH-}$$

is almost completely on the left side.