What is an official source that explains the rules that apply to the EEA in general (not just EU)?
The rules for the EEA are the same. The directive 2004/38/EC has been "incorporated into the EEA agreement." See https://www.efta.int/eea-lex/32004L0038.
According to the above-linked website, within the EU, applications for such a residence permit must be processed within at most 6 months. This is important since these residence permits expire 6 months after moving countries—thus a new one should be issued before then.
The six-month limit is not in fact particularly important because failure to have the card cannot lead to expulsion, only to an administrative sanction, and if the application is submitted promptly then any such sanction would be unjust. In fact, I have heard that in some places it is not particularly rare for the six-month deadline to be exceeded.
Also, for a family that has moved to a new country, the status of the non-EU/non-EEA family members does not depend on the possession of a residence card from the previous place of residence -- it is the same whether they have arrived from another EU/EEA country or from a non-EU/non-EEA country. And the terms under which the residence cards lose their validity are not necessarily uniform; this is governed by national law, subject to limits set forth in the directive, rather than by the directive itself.
(Note too that these cards are properly called "residence cards" rather than "residence permits" because they do not confer permission to reside; instead they are evidence that the bearer has a right to reside that is not subject to the host country's discretion.)
Are there similar rules on processing times for the EEA (not just EU)?
As noted above, the rules are the same.
In comments, you write
There are practical issues though such as being able to travel and return. Without a valid residence card, how can the family member prove the right to re-enter?
There certainly are practical issues. Without the card, the family member can use whatever proof of relationship would be used on initial arrival or to apply for the card itself. Such practical issues are certainly significant, and for certain purposes, for example obtaining employment, the card is an administrative requirement even if it is not a legal requirement.
We've been explicitly told that there is no guarantee of being let back in until the application is processed, for which the projected time is well over 6 months
There's no guarantee ever. For example, if the family member commits violent crimes showing that the family member is a threat to public safety, the family member can be excluded from the country of residence. But if the family member is armed with evidence of the family relationship, evidence of the EU (or EEA) citizen's citizenship, and evidence that he or she is traveling to join (or traveling with) the EU (or EEA) citizen, it would not be legal for the border authorities to deny admission without first finding that the family member is a threat to public safety, public health, or public policy.
Furthermore, in some places processing this application is a prerequisite to being able to access some basic things, such as opening a bank account, getting a medical prescription, a cell phone contract or even just public library access.
This is a significant problem. The solution will be different in different host countries (in part because the specific things that are difficult without the card will be different in different countries). The general solution is to complain to SOLVIT, where the EEA countries are also in scope.