False. 'Proxy wars' is a misleading title.
The Korean war was not a 'proxy war' it was a real war, that happened to also be a proxy for other powers.
Country A does not fight random country B for no reason.
They are generally real conflicts where their actions may happen to have geostragetic alignment with other powers.
No, they usually don't. But countries, nations, and people often have very long lasting disagreements. Instrumentalizing those, for another "bigger" cause isn't really that hard of a task for the far more influential and powerful countries C and D.
C and D end up supplying A and B with money, weapons, training and sometimes even direct manpower.
The result is usually a conflict that escalates much more quickly in scope and severity than it would have without the involvement of C and D.