There is no rule that salts don't react with acids or bases. This may be taught in some introductory classes, but if so, it is a huge oversimplification. In fact, it is a big enough oversimplification that we should probably just call it a lie. It is definitely something you should unlearn if you learned it.
Ammonia is a base. When you attach a proton, you get an acid (ammonium). So a salt like ammonium sulfate is not just a salt, but also an acid (though a very weak one). Therefore, if you later add a strong base to an ammonium salt, the base can steal that extra proton on $\ce{NH4+}$ and give back ammonia. This reaction tends to be obvious because ammonia is a gas, so it bubbles away. Ammonium is not a base, though, so it won't react with acids -- statement #1 from the question is false.
Salts can react with acids too. Let's take sodium carbonate as an example, formed from the reaction of sodium hydroxide and carbonic acid. The carbonic acid loses its protons and becomes carbonate, a base. It's a weak base, but it is a base. So if you add a strong acid to sodium carbonate, carbonate will accept two protons from that acid and you get carbonic acid back. Carbonic acid decomposes into carbon dioxide and water, so once again you get a gas (carbon dioxide) that bubbles away, and a nice visible reaction.
In fact, you can have salts that react with both acids and bases. Based on the previous two examples, you might guess that ammonium carbonate is one, and so it is. If you add a strong base you will get ammonia. If you add a strong acid you will get carbon dioxide.
So where does the original lie come from? Well, in many salts, the ions involved are so weakly acidic and/or basic that it's approximately true that they don't react with acids and/or bases. Sulfate is an example. It is a very weak base, though as a commenter noted, it can accept one proton to form bisulfate. So there will not be a notable reaction with an acid, certainly not one that releases ammonia. As long as you stick to using strong acids and bases to make your salts, you'll end up with salts where the ions are very weak acids and bases, and therefore are essentially unreactive to acids and bases.[1]
[1] Even this is a lie. Not all strong acids are equally strong, and other factors can cause reactions to occur. Sodium chloride reacts with concentrated sulfuric acid to produce HCl gas. The driving force in this reaction is the evaporation of the gas.
