Re: To feed for
I'm not surprised that you are a bit confused. If you have copied the passage correctly, there are a couple of errors which make comprehension more difficult. [First, however, remember that that the word "i" is always capitalised in the English language, so your " i would like to know" and "in fact i can't find" are wrong.]
"...even harder for Saudi Arabia to sustain the low prices that feeds its oil for security relationship with the United States." What is being fed is Saudi Arabia's relationship with the United States. The type of relationship is called an "oil for security" relationship. What feeds that relationship is the "the low prices" that Saudi Arabia charges. So, you have focussed on only part of the phraseology, hence your misunderstanding. The passage would have been much easier to understand if it had been presented thus: "...even harder for Saudi Arabia to sustain the low prices that feed its 'oil for security' relationship with the United States". [Notice "feed" rather than "feeds", to reflect the plural "prices".]
Now let's look at the second passage. This is not a well constructed sentence. Even I, as a native English speaker, had to read this a couple of times to be sure I understood it fully.
"And the United States will, in turn, have to adjust its Persian Gulf strategies since demands that regional GCC allies keep oil and gas prices too low, ironically, could create the very instability inside those countries that the United States so very much wants to avoid due to energy security considerations."
Who is doing the demanding? The US is doing the demanding. That part of the sentence would have been easier to understand if the writer had inserted a comma and an "its" as follows: "...adjust its Persian Gulf strategies, since its [or "US"] demands that regional GCC allies....".
The sentence is too long and too complicated for clarity; the "ironically" is in the wrong place; and the "due to" is used incorrectly. I could go on, but that's quite enough for now.
I'm not a teacher of English, but I have spoken it for (almost) all of my life....