Is enthalpy always negative?
The question s a little ambiguous, but if you mean the enthalpy change of a (spontaneous) reaction is not always negative.
It looks that way for most reactions at room temperature because room temperature is relatively low. Especially at higher temperatures you could have a positive enthalpy change if, in accordance with the Second Law of Thermodynamics, the reaction generates more entropy.
The formal equation for whether a reaction is spontaneous is
Note that the entropy change gets more important at higher temperature. Entropy change measures the tendency for atoms and molecules to assume a less ordered stare with more random variation, like a gas instead of a solid. Such more random configurations tend to have higher energy or enthalpy than more ordered configurations.
Let us look at the decomposition of limestone, primarily calcium carbonate, to lime (calcium oxide) and carbon dioxide gas. Like many important reactions, this has a positive enthalpy change but also a positive entropy change:
The enthalpy change is greater than zero because the solid calcium carbonate is more inherently stable than the solid but more reactive, strongly basic calcium oxide plus carbon dioxide gas. But the carbon dioxide gas has more random variation than the solid compounds because gas molecules can move freely through the volume of their container. So the entropy change is positive too.
Now look again at tgat free energy equation:
At room temperature the temperature is so low that there is not a lot of energy for forming the more disordered gas phase. The positive enthalpy change dominates the free energy and we do not get decomposition.