With that diatribe out of the way, I can get to answering the questions you asked.
Should we encourage posters to at least state their background?
In most cases, I don't think that posters would need to include their background unless it is very relevant to the question at hand. (The poster should be following the listed self-promotion policy if it applies). In fact, encouraging posters to include their background might cause an anchoring bias, where knowledge of the user's background prejudices the reader for or against their posted answer.
Should we be concerned about whether an answer is useful to the person who asked the question?
Yes, of course! This is a Q&A site, so useful answers are the main goal. However, I don't agree that encouraging posters to state their background helps to achieve this, as the usefulness of an answer should stand on its own merits and not need any background context to make it any more or any less useful. If the answer is about the wrong technology, or the wrong platform, or doesn't apply to the question, or overlooks important info provided by the question, then that answer should be treated as not useful because it is not useful rather than because it came from a user with background X.
To use some of your mentioned examples:
On Christianity.SE, a Catholic answer to a question asking for a Protestant perspective would actually be considered "not an answer" because no matter how well-written and sourced the answer is, it's an answer to the wrong question.
- (I'm not very familiar with this topic, so please let me know and I'll edit it out if I blunder into something offensive)
- Would an answer in a Protestant perspective be more useful if it came from a minister, from a long time practitioner, or from someone new to the Protestant faith?
On Math.SE, an answer involving calculus or other high-level math on a simple algebra question would likely be considered unsuitable.
- Would an answer based in simple algebra be more useful if it came from an algebra student, from an algebra teacher, or from someone with a PhD in Math?
On Stack Overflow, questions rooted in one programming language should not be answered with a solution in another language
- Would an answer in the correct language be more useful if it came from a programmer fluent in multiple languages, from a programmer familiar with only the mentioned language, or from a programmer?
It is often similarly unhelpful to try to "help" someone who has an issue with a program by suggesting they use a different program instead
- Would a solution in the mentioned program be more useful if it came from a regular user who had encountered and fixed the same problem, from a power user of the program who had dealt with it for years, or from an employee at the company which owns and supports the mentioned program?
Chances are that the answer from the more "expert" user will be a better answer than the one from the novice, but if they were otherwise the same exact content, does mentioning the source of the answer make one better than the other?
tl;dr
I think that we should not encourage the source of the answer to prejudice future readers about the content of the answer.
