Are aggregate roots always entities?

Yes.

The critical characteristic of an aggregate root is that it acts as a transaction consistency boundary.  In other words, it is responsible for changes that must always and immediately satisfy some business invariant.

That immediately rules out the possibility that it is a ValueObject, because ValueObjects are immutable.  Any changes to a value reflected in the model are going to introduce a new instance of the value object.

Similarly, DomainEvents are also ruled out -- events are things that happened in the past, and we don't have time machines.

As an aside, I think the DDD Sample gets this wrong; the HandlingEvent aggregate is modeled as a DomainEvent.  The description in the class header is

HandlingEvent's are sent from different Incident Logging Applications

Written in the active voice

Incident Logging Applications send HandlingEvents

and now I'm suspicious.  Does the business really not track the source of these external events?  Notice, also, that we never load a DomainEvent from the repository -- we only collect a history of events that match a TrackingId, which is a value object typically created within cargo.

DomainService is stateless, so there's no need for a transaction.

Sagas? I don't believe that's a fit; long running business processes that span multiple transactions and potentially more than one aggregate.

ApplicationService doesn't fit because the business invariant belongs in the model.