It is certainly appropriate to use "e.g." in a sentence without deploying parentheses. In your Example 2, I would place a comma before "e.g.". Parenthesizing examples is purely the prerogative of author, usually serving as a means of indicating the perceived importance of the examples to the thought being expressed.

It is certainly appropriate to use "e.g." in a sentence without deploying parentheses. In your Example 2, I would place a comma before "e.g.". Parenthesizing examples is purely the prerogative of the author, usually serving as a means of indicating the perceived importance of the examples to the thought being expressed.