In this particular case, the おう（ろう、こう、etc） or まい for a negative indicates a person's intention. So you are correct, in this context it is similar in meaning to ために or ように, although these two are more objective / place less emphasis on the internal thoughts of the person. と + a complex sentence indicates the way in which this intention is related to the action described after it.
The examples 1-3 you gave are just variants of this usage. All of them indicate a person's intention / thoughts and relate them to whatever they are about to do with the と or simply end the sentence there with した、思った、考えた etc if the sentence simply describes the intention or connects it to a less elaborate action. As you remarked, the usage in 2. is exactly equivalent to that in the first four sentences and the OPM quote.
To give an example translation for one of the sentences:
translates literally to
And then, one day, I called the principal of a certain school, intending to get him to let me get an audition.
or less literally
And then, one day, I called the principal of a certain school to get an audition.
Sometimes the volitional is literally used like a 'quote'. As an example, you could translate the sentence with the negative まい
"'I won't fall without a fight', he thought to himself, as he engaged the enemy, fully prepared to die in the process."
in which case you would be hard pressed to interchange the volitional with 「ために、」or「ように、」, since
would make it sound like survival was an important objective, while this is not the case in the original sentence. On the other hand, an alternative TL for the first sentence could be
"He engaged the enemy, fully prepared to die, in order to make his death count."
In which case you could reverse-translate it into
So there's certainly some clear overlap in the meaning, but they are not equivalent.
For the additional question (and to clarify a bit more):
I wouldn't say this sentence is the same as any of the other examples, but rather, kind of like a 'combination' of them. It could be interpreted as any of the following (which are not equivalent, so reverse-translation would not work):
I worked hard, intending to pass the exam.
Here, the intention is emphasized much more than in the original sentence. 気 emphasizes the state of mind, while つもり is more based on rational thought. I'd say both are equally close to the original sentence.
I worked hard, trying to pass the exam.
This one does indeed feel somewhat redundant. I can't say it's absolutely incorrect, but it sounds awkward, and usually you would drop the して. In terms of meaning, I think this would emphasize the idea that they tried doing different things in order to succeed. It doesn't sound equivalent to the original, in any case.
I worked hard, thinking to myself "I'm going to pass this exam".
While it's probably hard to tell from my translation, this emphasizes a combination of superficial thoughts and feelings. Unlike with つもり, this focuses less on rational thought and sounds lighter in tone. Unlike with 気, this is more shallow in tone (kind of like "simple thoughts" vs "a fundamental driving force"), and more thought-based than emotion-based.
For the OPM example, while grammatically, you could convert to any of the examples / all of them could be correct in a sense,
is the one that feels most natural (although not as natural as the original sentence). 重すぎる期待に応えるつもりで gives the impression that King was "planning" to meet the unrealistically heavy expectations (presumably based on some logical reason), while the speaker probably assumes that King's actions were more driven by an emotional basis (since he adds 焦っていかん). At least that's how I'd interpret it.
sounds like "I know your mind is going "I'm going to meet the expectations, no matter what." right now, but don't rush things, King." This is a very liberal TL of course. Still, I feel the original sentence did not put this much emphasis on thought, but rather focused on what King was about to do, so it does not feel as natural (to me) as the original or the one with として. It's possible that I'm interpreting this wrong, though. Either way, the ようと思って form is still quite close to the original, all things considered.
This is probably the type of thing that you get a better feel for in time, as you get more exposure to the language.