Should zero be followed by units?
This is actually a really interesting question.
In principle, "zero" doesn't need units. You can think of units as a multiplier - but multiplying zero by anything still leaves you with zero.
However, when you are talking about a physical quantity, it is very reasonable and appropriate to use units, even if the quantity is zero. And you have to use the correct units.
It's important to think about the situations in which it even makes sense to speak of "zero anything" - because the absence of a certain property has different implications in different situations. Think about this statement:
"The photon has zero rest mass" - in this case, there is no need to specify units. The mass is zero - it is simply a property that the photon does not have.
On the other hand, there are times where you are trying to determine whether something is really zero or not. For example, you might want to determine whether the charge of a neutron is truly zero. A careful experiment might conclude that the charge is $0 ± 1.234\cdot 10^{-34} ~\rm{C}$. The units are necessary - because while the number itself is zero, the uncertainty in the number is finite, and has units.
Finally, it is patently wrong to say "the neutron has 0 kg of charge" - which shows that although it is "nominally" the same as saying "the neutron has 0 charge", the units do matter.
Of course, in situations where the scale is arbitrary (that is, where 0 "units" does not correspond to the absence of the property) you always need to use the units. The example given in several of the answers of temperature (°C, K, F) is a good one. In general I believe this can only be true of intrinsic properties (that is, properties that are independent of the quantity of material).
As long as the quantity under consideration has a unit, yes, because of the importance of the Consistency of Units or Dimensional analysis. To equate, add or multiply quantities, they should be consistent. When one writes $y=ax+b$, the quantities $y$, $ax$ and $b$ should have the same dimension, i.e. the unit of $a$ times the unit of $x$ should be the same as the unit of $b$.
One should not add a unitless $0$ to a distance, while adding $0$ meter to that distance makes sense. Even if the quantity is $0$ "unit", I believe it still does matter with products, see xkcd: dimensional analysis.
When it comes to applying a more complicated function (a logarithm, an exponential) to a dimensionful number, the discussion is more involved, see for instance Exponential or logarithm of a dimensionful quantity?. Some advocate for instance that a logarithm is dimensionless (from What is the logarithm of a kilometer? Is it a dimensionless number?).
[EDIT] For real-world fans of dimension analysis, Why dimensional analysis matters by UnitFact:
The question cannot be answered generally because it depends on the situation - on what exactly you mean. If you mean "zero mass" then writing $0 \textrm{g}$ or $0 \textrm{kg}$ or something like that is very reasonable. If you mean an abstract, unitless zero from $\mathbb{R}$ - well, that one is unitless and should be written without units.
It strictly depends on what your numerical value wants to express. Numerical values wanting to express some physical quantity which has units should be followed by the appropriate unit, unitless quantities and abstract numbers should not be.