First answer to your specific doubt:
1. How the heat content of the system will change if temperature is constant : To change heat content , necessary condition is that there should exist a temperature gradient. That is to say your system can remain at constant temperature and still reject/accept heat if your surrounding is at a different temperature. Example is melting of ice or boiling of water. Both process reject/accept heat at constant temperature. There is however a temperature difference between ice/water system and its surrounding.
2. How the temperature of the system will change if heat content is constant : By work done on system. Hence temperature of system will change even if there is no heat content added or removed from system.
Please understand temperature and heat are two very different concepts.
Temperature is a state property like pressure, volume , internal energy etc and is used as a variable to define state of a system.
Heat is energy in transit which crosses boundary of system. It along with workwill drive changes in state properties like temperature. That is to say they together move system from one state to another, according to the given definition of problem.
Hence by definition of
Adiabatic : There is no heat transfer. Whatever work is done on system will drive change in state properties. Temperature being a state property may change and also pressure, specific volume , internal energy and other state properties as well.
Isothermal : There is no change in Temperature. There is however no restriction on drivers of change i.e heat and work. They can take finite non zero values and will drive change in other state properties like pressure and volume but not temperature(because of definition of isothermal process, and such process are physical possible for example melting of ice as I mentioned in the beginning).