How do I write out the chemical reaction of ammonia with strong acid?
In the second part, for example, $\ce{Zn}$ is 2+ and $\ce{H}$ is -1. So using the criss-cross method, you get $\ce{Zn(NO3)2}$ and water. I understand this.


I don't agree on the -1 for hydrogen, let's look again at that example:

$$\ce{2HNO3(aq) + ZnO(s) -&gt; Zn(NO3)2(aq) + H2O}$$

Metal oxides, such as zinc oxide, are basic anhydrides. 

Imagine that 


$\ce{ZnO}$ was made by heating $\ce{Zn(OH)2}$ until it loses water:


$$\ce{Zn(OH)2 -&gt;[\Delta] ZnO(s) + H2O}$$


$\ce{ZnO}$ is a salt of $\ce{Zn^2+}$ and $\ce{O^2-}$


The reaction between nitric acid and zinc oxide and just a neutralization reaction between two protons and $\ce{O^2-}$ (which, as such, isn't present in solution).

Now for the reaction with ammonia: For this you may want to remember that there are different acid-base theories: The one that helps here is the Bronsted-Lowry theory:


Acids are proton donors 


$\ce{HNO3}$ definitely is that.


Bases are proton acceptors 


How can something else other than the $\ce{OH-}$ that you knew and the $\ce{O^2-}$ from $\ce{ZnO}$ be a proton acceptor = form a bond to a (proton)? 

Is that possible if this acceptor doesn't even have a negative charge? 

(Yes, otherwise I wouldn't buld up the tension like in a suspense story :D)

If the proton can't provide the electrons for the bond, the acceptor has to deliver them! And that can happen in the form of an electron pair, a double filled, non-bonding orbital.

Therefore a reaction, such as 

$$\ce{H+ + NH3 &lt;=&gt; NH4+}$$

is perfectly valid. The rest of the example is just arranging the counter ions.