What do you call a part of a statement which gives away a bias?
Generally, this kind of work is called biased language or editorializing.
In your example 1, "quietly" is simply an adverb modifying the verb "passed". I know of no specific name for this rhetorical device.
Your example 2 fits the description of biased language or editorializing as stated by @FeliniusRex. The phrase "without evidence" can also be described as a parenthetical expression. Even though no parentheses are used, the phrase serves the function of adding information, but the sentence also makes perfect sense without it.