Does temperature remain constant throughout the isothermal process? Temperature is a state function, so it depends on the initial and final state of the thermodynamic process. So, does the temperature remain constant throughout the thermodynamic process? or does it remain constant only in the initial and final state but changes in between the process? 
Does the same go with the isochoric process? 
 A: Along a isochoric / isobaric / isothermal change of state the volume / pressure / temperature remains constant during the whole process. This can be achieved by coupling the system to a so called "bath".
A: The fact that the initial and final temperature of two equilibrium states of a system are the same does not necessarily mean the two states are connected by an isothermal (constant temperature) is process. It’s a necessary but not sufficient condition. 
The same applies when the initial and final pressure or volume are the same. They are not necessarily connected by an isobaric or isochoric process, respectively.
There are an infinite number of paths connecting two equilibrium states.

but if the temperature is the same throughout then only it is an
  isothermal process, isn't it?

My answer only addresses your question which, as I interpreted it, if the initial and final temperatures are the same is it an isothermal process. And the answer to that is not necessarily.
For a reversible isothermal process, the temperature of the entire gas is in equilibrium with the surroundings throughout the process. But as @Chet Miller pointed out, for an irreversible process the temperature of the gas is only the same as the surroundings at the boundary between them. Only in that sense is the process also called "isothermal". 
In any case, whether the process is reversible or not, having the same initial and final state temperature does not necessarily mean the process is "isothermal", in either the reversible or irreversible sense.
Hope this helps.
A: If the process is irreversible, "isothermal" usually means that the system is held in contact with a constant temperature reservoir throughout the process.  But this only guarantees that the boundary of the system in contact with the reservoir is at that temperature throughout the process.  Interior to the system, the temperature can vary with spatial position during the process, except at the final state where the system re-attains thermodynamic equilibrium at the reservoir temperature.
A: Isothermal means during the process you hold the temperature constant, so the temperature initially, during the process and at the end is the same. 
Same goes for isobaric and isochoric process as well. 
