Will the Heat Death scenario ever occur or just the Big Freeze? According to the Wikipedia page regarding the ultimate fate of the universe, heat death is a different scenario than the big freeze.

It states that in the big freeze, temperature will asymptotically approach absolute zero. However, the following section about heat death states that heat death may only occur if temperature reaches an eventual temperature minimum.

If temperature never truly reaches absolute zero (the temperature minimum), can heat death even occur in the first place?
The Wikipedia article is somewhat confused on this point (at least one of the references does not support the sentence it is supposed to support). 

The classic concept of the heat death of the universe was that eventually it will run out of free energy and there will hence be no possibility of any work. A more modern view of heat death is an approach to maximum entropy where the system will be in equilibrium. This does not rule out random fluctuations bringing it away from equilibrium, but they are so rare it does not matter. 

Note that it is entirely possible to have a heat death at any temperature. 

The big freeze would happen if expansion of the universe led to temperatures decreasing towards absolute zero. However, if there is accelerating expansion this generates a constant finite temperature horizon radiation, about $10^{-29}$ K. This means that the big freeze will never happen. 

Had there not been horizon radiation it would in principle have been possible to avoid heat death by storing some background heat in a perfectly reflective box, waiting until the universe become colder, and use it to perform work. This can be repeated indefinitely. (Not really: the timescales quickly become long enough that random fluctuations destroy your device, but that it a separate issue).