there is a mechanic for this actually, where GF summon-speed speeds up or slows down depending on which spells you cast? I don't really know
This is indeed the case.
Casting a GF increases compatibility (for the specific character doing the summoning only, obviously) with that GF by around 20 points, the compatibility with the opposite GF decreases by 10, and the compatibility with every other GF decreases by 1.5. There is one exception to this, who is the KotR-equivalent GF; summoning that one increases the compatibility with it by 2, and the compatibility with every other GF by 1, which is a strange dynamic. Also, not every GF has an opposite; Diablos, for example, since he's not-elemental. In those cases, it's just the +20 to itself and -1.5 to everything else.
For reference, Ice opposes Fire, Thunder opposes Water, Earth opposes Wind, Poison opposes Holy, and Negative-Status-Imposing opposes Positive-Status-Imposing, which should allow for instantaneous recognition of how to not pair GF without spoiling anything.
Casting spells also increases compatibility: ice spells increase compatibility with Shiva, while Fire spells decrease compatibility with Shiva; both of those work in reverse for Ifrit, naturally. Interestingly, Thunder and Wind spell both increase compatibility with Quetzacotl, but only the Wind spells increase compatibility with the Wind-type GF; and with Siren, every status inflicting spell increases compatibility, with Silence doing so by the largest amount, but using either Esuna or Dispel decreases the compatibility. And casting Ultima will generally increase the compatibility with every GF. In general, the idea seems to be to cast spells the GF would approve of, although, even in the most positive cases, such as Blizzaga for Shiva, the compatibility increases aren't that high (+2.5), and in the weaker cases the increase or reduction is meaningless (+/-0.2), so it isn't really something to think overly much about.
All this explained... Compatibility is of very limited meaning. It does matters, in that a 1000 Compatibility GF (that's the max) will have its charge time be just 2.8 seconds, while a 0 Compatibility GF (the minimum) takes 16.8 seconds to charge - that can be relevant when deciding which GF to attack with and which ones to use as HP buffers. But overall, the system has very limited importance, so I agree that having done more with that, even just on a narrative level, would not have been a bad idea.
I'm not 100% sure, and I'm not able to find any confirmation in the spoiler filled guides, but I'm fairly sure that you have to pay for this map... unless Rinoa is in your party.
I checked, and this does not appear to be the case; I got the map for free when the team was Rinoa and Irvine, and also got it for free when I went with the Quistis-Irvine combination. This suggest to me that the map is always free, although, as mentioned above, the dialogue without Rinoa is different; in Quistis case she says "that's an insult to SeeD" when they're ordered to go do this busywork of retrieving the code that was clearly left there on purpose.
I suspect that the whole sequence is just a flex from Caraway, that he's "putting these teenagers in their place", to show that just because they're superhumanly powerful doesn't mean he can't force them to do whatever he wants, and the whole "disappeared student" is an excuse.