A democratic republic is a country which is both a republic and a democracy. However, in practice countries which describe themselves as democratic republics do not always hold free or fair elections. An example of this was the German Democratic Republic, a communist state commonly known as East Germany.
The GDR may have described itself as a democratic republic, but that was a pretense. In reality, it was an aristocratic republic -- which is also true of the United States, but for a completely different reason.
A constitutional republic is a state in which the head of state and other officials are representatives of the people and must govern according to existing constitutional law that limits the government's power over all of its citizens.
This is a wildly idiosyncratic definition. By any normal understanding of the term, a "constitutional republic" is simply a republic whose operations are prescribed in a written constitution. You have incorporated elements of democracy (officials being "representatives of the people") and protections of rights ("limits the government's power over all of its citizens"), neither of which is a necessary or sufficient condition for constitutionality.
In fact, "constitutional republic" is a redundancy. There is no such thing as a non-constitutional republic. The important distinction among republics is whether they are aristocratic or democratic, not whether they are constitutional, because all republics are.
Because the head of the state is elected, it is a republic and not a monarchy.
No, that doesn't follow. The Kings of England prior to the Norman invasion of 1066 were elected by the Witenagemot, a precursor to Parliament in which the nobility voted. Election of the monarch was a fairly common procedure, about as common throughout history as hereditary succession.
What distinguishes a republic from a monarchy is that under a republic no one person holds absolute rule. A republic is collective government, at least to some degree. It may be that only a privileged few can exercise real political power, but it is never a single person.
In a constitutional republic, executive, legislative, and judicial powers are separated into distinct branches.
Yes, this is often true -- or rather, in a republic, that is the case. (All republics are "constitutional.") However, it's not absolutely true. What is absolutely and always true is that in a republic no one person holds the executive, legislative, and judicial powers by himself. Power in a republic is always shared.
The fact that a constitution exists that limits the government's power makes the state constitutional.
No. The constitution prescribes the government's functioning; it does not necessarily limit the government's power.