Is the reason behind the cheapness of one unit of Japanese/South Korean currency compared to most other currencies the high inflation in WWII for Japan and in the Korean War for Korea?

No.

The Korean won and the Japanese yen were both pegged to the US dollar at a fixed rate for some time - the yen until 1973 and the won until 1997/98. Prior to these dates, the rates were adjusted periodically, but the value of the yen/won always fell in relation to the dollar. You can study basic economics, namely supply and demand and export-driven economies to understand how and why this happened, but the short version is that an export driven economy does what it needs to do to make their currency weak so they can sell sell sell.

As far as numbers and par goes, the US has dollars and cents. Japan and Korea have the yen and the won, respectively. They also have smaller units, but those are antiquated and not used. That said, today, 1 USD is 109 JPY, or 100 cents is 109 yen, 1:1.1. The USD to KRW is 1 USD to 1183 KRW, or 1:11, a difference of 10x. If you use the deprecated values, the jeon is 1/100 of the won, so you could say that the exchange rate is 118300 jeon per dollar. The point is that it's arbitrary. The USD is worth 32 Taiwanese dollars, or 7.76 Hong Kong dollars.

It is true that because of World War II and the Korean War, the respective currencies lost a ton of value, but that is irrelevant. The currencies regained value after the war, they were revalued a few times, and with a floating exchange rate, the value can (and has) rise and fall on any sort of economic news.