Are all NO3- salts soluble in water? If so, why?
No, the correct way of putting it is
$$\mathrm{Almost~all~of~the~\mathbf{inorganic}~nitrate~salts~are~soluble~in~water. }$$

The families of organic nitrate salts are typically nitrates of azoles and imidazoles. Some bright examples are (R) &amp; (S)-miconazole nitrates, isoconazole nitrate and econazol nitrate. Econazol nitrate (Other names: Spectazole, EN) is the most common organic nitrate salt, and is believed to be very slightly soluble in water, less than $0.1 \mathrm{\frac{g}{100~g}}$ (1, 2, 3)

In inorganic nitrate salts, with a bit of patience, I've noticed that barium nitrate, mercury(I) nitrate, and cobalt(III) nitrate are the least soluble of nitrate salts with a monoatomic cation. They are in the brink of insolubility, but I still wouldn't call them insoluble, since


Their solubility isn't that low at STP; somewhere around $\frac{5~\mathrm{g}}{100~\mathrm{g}}$. ($\ce{Hg2(NO3)2.2H2O}$ decomposes in water, so as RSC says)
Their solubility increases greatly with temperature.




The reason, you say?

Well, let's think of why one compound is soluble in water while the other isn't. When an ionic compound is formed, energy is released. This is called Lattice enthalpy in constant pressure. The process of formation of lattice from separate ions is usually highly exothermic. Therefore, when an ionic compound dissolves in water, and when this lattice "breaks up", the reaction would be highly endothermic. 

So, to dissolve in water, the ions have to 'overcome' the lattice energy. How? Of course, the net resulting attractions between water molecules and the ions have to be stronger. This attraction is kind of a formation of a very very weak bond (and its weakness is the reason it's not categorized as such) and thus, is exothermic. This energy release when the dissolution occurs is called enthalpy of hydration, provided that the ions be in the gaseous state.

So, nitrate is a really big anion, with a single charge. The lesser concentration of negative charge results in relatively lesser lattice enthalpies. Also, the possibility of the creation of hydrogen bonds with water enhances solubility by increasing enthalpy of hydration.

That's why almost all of nitrates are soluble.