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In a phrase like:
面白{おもしろ}半分に見る
or
面白半分に言う
What is the concept behind the 半分 and why is it connected with the verb using に?
Is it like "all things have two parts/aspects": serious (真剣半分?) and fun and the action takes part "on the" fun part (as opposed to the other equal part), thus being similar to the English "side of the coin"? Or is it something different?
Are there any other phrases using this concept?
The 面白 comes from 面白い which as you probably know means "fun, interesting". The 半分 part means "half".
The expression ”面白半分に” means literally to do something "half fun", "half serious" (as you hinted at). See the definition in the dictionary here. The "に" acts to make the phrase an adverb which is acting on a verb such as 見る or 言う in your examples.
The Japanese definition of this word says "興味本位の気持ちがあって、真剣さに欠けていること" which means something like "Having a curiosity about something, but lacking in seriousness".
I think this can be translated in various ways depending on the situation, such as "just for fun", "half in jest", etc.
You can see this page (Japanese) which gives a brief description of the phrase. The author of that article indicates it may have a slight negative connotation.
