So what happens when this now aggressive species applies for membership? Will the Federation give it to them on the condition they don't conquer anymore? Hey, they've got their goal, which is to get all this juicy tech. So now they have photon torpedoes, phasers, anti-matter, basically everything you'd need for a population conquered and pissed-off at their invaders, thus ensuring a healthy resistance movement. And so now what happens when portions break away and form new nations? Does the Federation interfere by sending soldiers and troops? Or does it kick the member out for no longer being united? If they get kicked out, doesn't this just leave the original superpower to grow angry and possibly commit genocide to wipe out everyone who might possibly get in the way of their having advanced technology?
The episode
The Hunted indicates that, no, your scenario would not play out as such. The Federation evaluates applicants based on more criteria than just "one government", and has in fact rejected applicants which had experienced planetary unity for many years. I don't have the episode handy so I can't tell you what those criteria are, but I'm sure key factors include the proportion of children successfully processed by communist boarding school indoctrination camps, as well as the degree to which personal property ownership is stifled.
Additionally, the development of warp drive is the point at which the Federation can no longer control whether or not the less advanced civilization encounters aliens. That's the reason why contact becomes permissible at that point - because contact has become effectively inevitable. Better to do it while at least one side is expecting it and has prepared for it as best as they can.
The Klingon Civil War example you give doesn't entirely make sense considering that Starfleet and the Federation Council approved the dispatch of an expeditionary force* into Klingon space in order to detect and deter Romulan intervention. This was in fact a form of Federation intervention, as the Romulans would have gotten away with their plan to supply the Duras insurrection had Picard not gotten involved.
(Said expeditionary force was, despite Sela's derision, a fairly substantial commitment on the part of the Federation, considering that it represented the
total effective Starfleet strength in that area.)
In fact, this approach turned out to be the
better choice to make, as simply sending the fleet in to support Gowron could have led to a protracted conflict wherein the Federation was indirectly fighting the Romulan flow of supplies. Exposing the Romulan connection effectively destroyed Klingon support for the insurrection far more quickly and with much less damage to the Federation/Klingon strategic position against the Romulans.
Additionally, your "damaged starship" scenario is contrived at best. "Oh, it's really badly damaged... just enough to take out
all of their FTL communications (and those on the shuttles...

) but not so much as to totally wreck or destroy it (or damage their transporter system), and it happens right next to a conveniently populated planet that just happens to have "supplies" that would be super helpful, if only it weren't for those damn liberal big gov't regulations!" The odds of such a scenario occurring are vanishingly small, so much so that it should not typically enter consideration. What kind of supplies are they going to trade for? What are they going to trade for those supplies? How disruptive is that trade going to be?
Let's say you decide you're going to exchange medical technology/information for... fuck, I don't know what we could offer a starship, let's say a few thousand tons of titanium. Who do you trade with? Suppose they start a war over who gets to trade with you. Or suppose whatever you trade them gets used by whomever you trade with to gain a strategic advantage over their neighbors.
This is not to say that the Prime Directive is always right and must be adhered to blah blah blah, or even to try and justify unnecessarily constrictive writing practices like Michael Piller was prone to do. Rather, I'm trying to point out that you might be looking at things a bit simplistically (and somewhat histrionically). You're right, as a 'sacred and inviolate rule' it stinks, and some of the writers have gone waaaaay overboard in promoting it - but a general principle of non-intervention is still not a bad idea in and of itself.
First Contact is an episode highlighting the negative attitude the writers had for the military, while thinking that by sticking to their Prime Directive bullshit, everything would work out in the end because everyone would just shrug their shoulders and say "oh, that makes sense." Imagine how angry someone would be if the Federation shows up and offers advanced medical treatment, but they've been watching for over a year and could have easily treated a family member's disease from which they died. Are they going to shrug their shoulders and say, "Oh well, they knew what they were doing."
Also, why did that scientist only need to go up to the minister to request funding, instead of some sort of parliament or congress, some lawmaking body, to request funds? Does the minister control the purse strings of the entire government that he can fling around on a whim? What about annual budgets and research grants? Is he space Obama? Is that how things work in Star Trek? Just go up to someone and ask for money, since there is no money? It'd be like saying, here, have a few extra oxygen molecules, I don't need them.
This scenario already occurs every day. Throughout the world are a number of tribes of human beings living in extreme isolation. They are almost totally ignorant of the outside world and of nearly every technological concept we take for granted, including the vast majority of medicinal knowledge. Attempts to integrate them into our society have always been disastrous. This is not to say that contact should never be made, but that there is legitimate cause to be extremely cautious in initiating contact and not for anything so brutishly simple as 'well we got some great stuff here let's just go give it to 'em so they'll live better'.
Your last remarks seem almost purposefully obtuse. To my knowledge the only space nation explicitly stated to have abandoned money is the Federation, so whining about the money issue with regard to a civilization that just barely built a warp drive seems out of place. In a wholly self-contained 44 minute story, it simplifies the script considerably to reduce the number of people, places, and events that the audience has to keep track of. And besides, it's enormously cheaper to just have the two already-budgeted guest stars talk to each other than to try and get something as large and unwieldy as a parliament portrayed, especially when we're never going to see it again. It's okay to take television production realities into consideration when you're thinking about an episode, it really is.
But seriously, dude, that was a lot of words. Did Gene Roddenberry sleep with your wife?
(Also a bit of pedantic trivia: Roddenberry probably wasn't really involved with the episode First Contact. At that point in the series he was pretty ill and did not participate much in running TNG. He may have had brief plot blurbs run by him for his approval but he certainly was not editing or rewriting scripts anymore.)