To answer your first question, yes two (different) adiabatic paths can intersect in a plot. See State A in Fig 2. However, for this to be possible, at least one of them has to be irreversible. For different reversible adiabatic processes involving the same working substance, their plots cannot intersect. See Fig 1.
Now, if you could have two reversible adiabatic processes intersect and be connected by a reversible isothermal process, there would be a violation of the second law. In that context, the answer to the linked question where both adiabatic processes are shown as reversible, would be correct. That is because both the system and the surroundings would return to their original states with no increase in entropy.
However, since at least one adiabatic process must be irreversible, then the cycle A-B-C-A shown in Fig 2 below would not violate the Kelvin-Plank statement of the second law. Recall the law states:
“It is impossible to construct a device which operates on a cycle and produces no other effect (my emphasis) than the production of work and the transfer of heat from a single body”.
The irreversible adiabatic process (Process A-B) in Fig 2 produces an increase in entropy. Thus cycle A-B-C-A does constitute some other effect than the production of work and transfer of heat from a single body, and therefore does not violate the Kelvin-Plank statement of the second law.
Hope this helps