For instance, if the wording states "protect me," could one add "me and my family?" Or if the wording refers to "we" (plural) could it be modified to "I" (singular) or vice-versa?
Regarding the main prayer - the Amidah.
Yes and no. Prayers are two-fold, they have two [contradicting] aspects (as per Gemmorah in Brochos 26b):
- They are personal requests
- They reflect/replace public offerings
Following the first approach, we'd say that a person is free to modify the text according to his personal needs. Following the second, we'd say the phrasing is fixed and unchangeable.
The Halachic resolution is that most of the prayer is left unmodified and is recited exactly as written, but some additions are allowed at certain places (in Rafaenu or before Shomeah Tefilah. See the links in comments).
Regarding other prayers and Brochos - there are different levels of exactness for different parts of service, but all the requests that are not exact verses are not forbidden from being altered BUT based on one's needs, not interpretations.
For example, one can add a list of additional requests after Amidah but one can't render the request for building the Temple irrelevant for him.