There seem to be two basic issues in this question. The first is similar to the issue of when a pronoun's antecedent is clear; is the extra information necessary for an understanding of the basic meaning of the text. This is addressed by Lauren Ipsum's answer: "if there is no other reasonable interpretation" when the extra information is excluded, then "you can take it out." As with pronouns and antecedents, this may be less than obvious to the writer in some cases since the writer knows what was meant, but typically an analytical reading can make the determination.
The other issue is whether the extra information serves a function other than expressing the basic meaning. Extraneous information can be used to provide the desired length or rhythm (e.g., for a slower pace), to communicate in sound or connotation some addition information, to express some effect from the level of detail itself, to provide a foundation or link for adding information, to make a subtle point, or even to include some element of repetition that is significant to the theme of the story or to one of the characters.
Use for length or rhythm
While this is generally not the best method for adding length, it can be acceptable, particularly when other purposes carry part of the weight of the extraneous information. In the first example, "to the ground" would seem to be acceptable for a slower pace if the narrator is casually considering death (in which case the more casual "nearly" might replace the more analytical "approximately" and a comma replace the period after "fall"), especially with the sound/connotation effect of ground (explained later). In the second example, the pacing is presumably meant to be fast so extra syllables would be counterproductive. (Also "pinched" might be a more precise and accurate verb than "covered", having the side benefit of a harsh sound and sharp connotation, with sound similarity to "pain", "punch", and "pin" and a sense of squeezing tightly to the point of mild pain, which fits with the olfactory "pain" caused by the stench. It also has one less syllable, further improving pacing and sharpness.)
While extreme attention to rhythm is more significant for poetry, in some cases prose can benefit from changes that allow continuing a regular beat or disrupting the rhythm. This is closely related to pacing (length and the rhythm of natural stresses both influence pacing), but can also present subtle effects like implying order with a regular rhythm or strengthening a transition or a "harsh" concept by breaking an established rhythm. In the first example, "'ted my flashlight down to the ground" scans as ˘ ˘ ´ ˘ ´ ˘ ˘ ´ where the intermediate iamb, nearly a spondee (´ ´), "'light down" almost gives a sense of the tension of tripping with the following anapest (˘ ˘ ´) perhaps hinting at a short fall (two unstressed syllables) and final impact.
Use of the sound or connotation
In some cases, the denotation of the words is not essential but the sound and sense of the words adds worthwhile meaning. In the first example text, "ground" has a harsh sound and a hard connotation, perhaps hinting at the impact from a fall. This effect is somewhat enhanced by its following the somewhat repetitive (and mildly oppressive) in meaning and sound "down".
Extensive use of such effects is less appropriate for prose (where sound and connotation are expected to be used more subtly) and would be jarring if noticeable. However, used judiciously, such effects can provide some of the emotive effects associated with poetry.
Use as unusually extensive detail
Including seemingly extraneous information can present an effect from the level of detail itself. This might parallel the pacing use in presenting a calm or bored tone ("counting flowers on the wall"), but it can also express extreme observation (à la Sherlock Holmes) or a consciousness of specific aspects (e.g., attention to a person's body). For the latter, this would typically be joined with adding further information. For example, if the narrator particularly notices Cath's bodily features (e.g., romantically or in comparison), «Cath turned her head, her petite hand holding her nose, "Oi, what a stench!"» states not only that the narrator notes that her hand is small (perhaps cute, reinforced with "Oi" and the slightly softer sounding "stench") but also that attention is paid to her physical form. (The slower pace of that sentence would further emphasize the narrator's attention to Cath.)
Excessive detail can also be used as distraction, by casually drawing attention to a feature the reader can be deceived into believing that there is some significance to that feature. This drawing of attention may also make other information less noticeable.
Use for adding information
A word or phrase which is unnecessary by itself can be used to allow adding information, whether by description of the extraneous component or providing a link that naturally flows into the additional information. For example, «Cath stopped her nose with her fingers, revealing her ring, "Woo! What died in here!"» uses "with her fingers" to mention the showing of the ring where "stopped her nose, revealing her ring" is less clear. (Incidentally, that phrasing could be used to imply the narrator feeling, after the revelation of the engagement ring, "my hope for a romantic relationship [is what died]".) Similarly, "Cath pinched her nose with her bleeding hand" allows the addition of "bleeding" (perhaps she cut it earlier and only then is revealed to be bleeding).
Use for subtext
This is somewhat related to providing additional information with the distinction that the information is not explicitly declared. For example, "Cath pinched her nose with her left hand." might be used to subtly remind the reader that fingers in her right hand were in a splint. Alternatively, the use of her hand, although natural, might be a deviation from her expected lady-like use of a handkerchief; this would mildly reinforce her atypical use of coarse language (she did not say "My word! What an unpleasant odor!").
In the first example text, being able to see the ground might be used to indicate that no awnings or other obstruction would break a person's fall. This lack of obstructions need not be unusual to be noteworthy, but it would require that some preceding (or possibly merely proximate) statement provided the appropriate context. (For example, in the previous chapter, the narrator might have been looking out a high-rise apartment window and see "the awnings like cheerfully colored firemen's nets stretched out from the bottom of the building"; this might even be used in that portion to express an urging to jump or to foreshadow such.)
Use of repetitive concepts
Unnecessary content can also be used to reinforce something significant to the theme of the story or to one of the characters. This is effectively a broader and probably lighter-handed use of subtext and connotation. By using unnecessary words that consistently point to a particular idea or feeling, the idea or feeling can be subtly established.
Because the words are unnecessary, their inclusion implies an unknown significance. This can be used as a background sound that the reader may only notice at the end or may never consciously realize is contributing to the emotional effect of the writing.
Explicitly stating "with her hand" might be one of many mentions of that character or people generally using their hands ("she sprinkled on the cheese with her fingers", "she grasped the knob in her hand", etc.), foreshadowing a reveal that the narrator has a crippled hand (and these were all reminders of what was lost) or that the character will lose the use of her hand (reinforcing the tragedy).
Similarly, "ground" could be used as one of many references of varying subtly to burial (and thus death).
Obviously, in a shorter writing, such cues can be inserted more frequently and with somewhat less subtly without the reader becoming fully aware of the intent before the writer intends. It may also be possible in a longer work to introduce a Pavlovian response to particular concepts. This would be an effect similar to background music but likely with less benefit from a common human or cultural association of concepts with the desired expectation. Like background music, the intended effect can be lost if it becomes noticeable.
In addition to asking if a particular phrase is necessary, a writer can ask if it is useful. In non-fiction, extraneous content generally works against the desired clarity and conciseness, but in fiction (and poetry) clarity and brevity may be less important than other considerations such as emotive effect and may even be contrary to the goals of the writer.