Don't worry about it, seriously. There is nothing awkward about being poor and it definitely is no reason to feel guilty, it just sucks. You've probably also experienced that people won't necessarily help you out under these circumstances, even if you specifically ask for help and if you know they have the money. So, dignity is all you have going for you.
Also remember, there is wealth in other things as well besides money. Besides buying rounds when I could, I've always made a huge point out of volunteering and helping people in other ways. Throw your time, smarts and skills around like some people spend their money. Learn how to cook. Learn how to repair stuff. You'll be fine.
True story: I have two best friends that I've known since kindergarten. When I was in a really difficult period a couple of months ago one of them said to me: "You know, if you really get in to trouble, I will always come and bail you out." We've never had to try it out, but you can't imagine the feeling of support that gave me. Now, I've just recently discovered that my other friend is actually insanely rich (compared to all the rest of us), he just never told us before. Guess who typically buys the rounds? Right, the first friend. Guess who is happier? Yup, still the first guy. But, I still love them both. Life is like that.
I don't think this is a true (or healthy) approach. Consider helping a total stranger who you can be fairly certain off you will never see again. E.g. an old lady having trouble loading her groceries in a city you're visiting as a tourist. (To keep it simple.) By your rational (long-term account balance between you and the lady) one should never help her! This seems like a pretty depressing approach to life.
Secondly: what about situations where there is a potential for a long-term friendship, but you don't know yet? I often make good friends just by helping them out in a pretty intense way right off the bat. (E.g. someone new moves to my town and I help them find a place to live, happened many times.)
No, I think a "flip the roles" system works better to judge these types of deals: "If I were an old lady in that situation, what would I expect?" "If I was new in town, how would I feel if someone were to help me out? What would it mean for me in the long term?"
Also keep in mind that it's frighteningly easy to pick out friends who have a "long term balance" system in their heads and I think it reflects badly on them. Don't be that guy.
When "flipping the role", it is important to keep all the parameters identical! In other words: you have to be able to empathize enough with the other person to be able to incorporate their needs/wants/capabilities in your "mental model" of them.
A practical example: when I'm having beers with someone who is studying (and very low on cash because of that), I very frequently pick up the entire tab. Why? Because I can still imagine what it was like as a student myself and sort of even imagine that that person will hopefully do the same when they eventually start making money. Vice versa, I would never expect them to pay more than "their own consumption" and would flat-out refuse to let them pick up an entire tab. (Unless there is an obvious indication they are swimming in money.)
This is basically what empathy means: being able to incorporate a workable mental model of a different person and being able to think from that perspective. Of course, there are cases where you have no clue as to the other persons background/situation, etc. In this case I'd suggest expecting as little as possible, because it would be easy to go wrong there. But you can avoid this situation by being observant and by asking about them. A simple "how are you" is a good start.
OK, I'll try and answer with fairly straight-forward points, as you seem to be in need of practical ways of judging these kinds of decisions:
1) When thinking about "things that need changing", it is very important to know two things first: a) How important is it to you? b) How easy is it to obtain?
You then decide if or not the combination of the two surpasses your "internal threshold". Or, if you're so inclined, think of it as an equation: if (difficulty / importance) > threshold: go for it.
In human language: If something is important to you, but it is extremely/impossibly difficult, you might want to just let it go. This is basically what the prayer alludes to.
Of course, you realize by now that the wisdom actually lies in being able to judge importance, difficulty and your internal threshold. This is something which you will simply have to practice all your life. If you keep yourself mindful about what worked/what didn't work and what made you happy/what didn't, this is doable.
Another helpful tip in this area is to realize that your idea of what is important is inherently subjective. Most philosophical thinkers will then tell you that you should tune down the importance of "need" (for physical stuff especially) and tune up the importance of moral values (e.g. loyalty, etc).
2) This one is fairly easy to solve: flip the roles in your head. What if someone were to ask you in exactly the same circumstances for the same thing? What would you feel about that? This should give you guidance in the appropriateness of the question. (This is by the way pretty much the "do unto others as you would them do unto you" rule.)
3) Productivity that simultaneously makes you unhappy while doing it and provides you with long-term happiness is as undesirable as non-productivity ("comfort") that makes you happy in the instant and unhappy in the long-term. Why not go for productivity and non-productivity that makes you happy both in the instant AND in the long-term? None of these things are mutually exclusive. Breaking this down is fairly easy: do a job you enjoy (no excuses!) and give yourself well-deserved breaks when you've earned them (no excuses). Neither of those situations should make you feel guilty in the slightest, otherwise you're doing something wrong.
Psychologically, consider this: in the long-term, you will only remember the fun and exciting bits (both work-wise and leisure-wise). You will not even remember the "hard but boring work" in a few years! This implies that your long-term happiness relies on being able to aggregate whole series of happy memories, none of the boring bits are important.
Caveat 1: Sometimes you do have to do something you don't enjoy. Just don't make it a life-style. Caveat 2: Some "comfort" gives you as little positive memories as boring work does. I'd put doing drugs and playing a lot of video-games in that category, but you can make up your own mind about that.
Your idea of "otherwise-sane people" seems off to me. "Sanity" is not a binary equation, a switch that you can flip on or off. In fact, you're probably well aware that extremely rational people are susceptible to a lot of the same psychological effect as "the rest". And there is no reason to think that awareness of such effects as mass hysteria is a full-proof shield for it.
In any case, on to your question. A room full of hardcore scientists can still easily be fooled by a skilled magician and if they are not told they are dealing with a magician, they can even be fooled into a "shared vivid hallucination". Worse still, in all likelihood some of these hypothetical scientists and "otherwise-sane people" might be the most prominent defenders of their "observations".
Good point. It took me some time to rethink the relevant strategy.
OK, so a workable strategy might actually be a slight variation on the proposed system. Let's head back for a second: why do we sometimes feel uncomfortable requesting something?
I was thinking about this and I believe it might come down to our dread for getting a "no". In a sense, when people respond positively, there seems to be no problem with the appropriateness: it was appropriate to ask*. It is only when you get a reluctant "maybe..." or a "no" that things get really uncomfortable.
This changes the problem into a matter of being able to predict the outcome of the request. In other words, being able to predict the answers that people are going to give. If you can be fairly certain you'll get a yes, it is most likely appropriate to ask. This skill will invariably require empathy and (yes!) a "simulation of an external mental state" (aka "flip the roles").
*: Mind you , you'll also have to mix in moral considerations, because some people are easy to take advantage of this way. (When people are "unable to say no".) Luckily that is a pretty rare situation.