Question: I'm in high school and wanted to get a few things cleared up. 

Isothermal process is defined as a thermodynamic process where temperature remains constant. 
Does this mean that temperature remains constant at every instant? Does an isothermal process have to be reversible?

The Joule expansion of an ideal gas is an irreversible process where there is no net change in internal energy. Can it be called an isothermal process?
If yes, is this the only way an isothermal irreversible process can be realised? What are other scenarios where such a process can be done? (Consider an ideal gas)

Answer: 
Most people consider an isothermal irreversible process as one in which the system is held in contact with a constant temperature reservoir at the initial gas temperature throughout the process. This says nothing about the spatial and temporal variations in temperature interior to the system, only at its boundary with the reservoir. Even in the Joule expansion, except at the beginning and end, there can be temperature variations within the gas.
An isothermal process is a thermodynamic process in which if you take to measure the temperature of the system at some time in the process and then measure it again, later in the process, the measurement of would be the same, independent of what two times you decided at whim to measure. 

Reversibility is a completely separate question. It is a question you ask in the context of entropy. If the entropy change of the system and surroundings for some process is zero, then the process is labeled reversible. Physically, this means you can retrace the whole process to the original state i.e: the whole process is a collection of many successive equilibrium states. You move from one equilibrium state into the next then the next and the final state of this chain of equilibria is your final state.

For an ideal gas, it turns out that

$ U= n C\_v T$

That means that the internal energy is only dependent on the temperature and not on the volume or pressure. Having an expansion of gas does not change it's energy, to change energy the gas either needs to expand against a pressure gradient or it has to be in contact with a body of a different temperature ( see zeroth law of thermodynamics)

As per other isothermal processes part, I don't know of any except the free expansions but there could be. I'd be grateful if someone could complete this part of my answer.