When measuring the temperature of a given system, all the thermometers give the same results.
Ideally, it is true. As long as the system is in thermal equilibrium with itself (no heat enters or leaves it, and it is allowed unlimited time to settle under its own intrinsic characteristics), the average kinetic energy of all the molecules wouldn't vary throughout the system. Hence, different thermometers should report the same temperature. Read more on this here.
When a system goes from thermodynamic state 1 to state 2, the variation of its internal energy depends on the process that originated the change of state.
No that cannot be true. This just follows from conservation of energy. Internal energy is the sum of kinetic and potential energy for all particles inside a system. There are a special class of functions which do not depend on the path taken by a system to reach the final state and are called state functions. You can arrive at why is internal energy a state function while heat and work are not by simply applying First law of thermodynamics.
In simpler terms, to get from state $A$ to state $B$, the system gains a net amount of energy $\Delta U=U_B−U_A$. It does not matter what the process was to achieve the state $B$. Of course, the total amount of energy transferred through different methods has to be the same $\Delta U$ due to energy conservation.