Honestly this is probably how it would go:
"Hi, my laptop won't turn on. I've tried replacing the power cells, charging it, but nothing's happening. Can you help?"
"Hmm... have you turned it off and back on again?"
"Umm... it's not turning on."
"It's turning on?"
"No, it's-"
"Hooray. I am smart. I am strong. I will go."
"Wait, don't hang-"
(hangs up)
That's true, TWOK requires him being on Enterprise for Season 1, or at least Space Seed. I wouldn't hold stardates too closely since they basically pulled them out of a hat and used whatever was pulled. But he would be in his late teens.
A cliffhanger... I was hoping they wouldn't introduce those. I expect Sam and M'Benga will be the only one who makes it out alive... expecting to see Sulu and Chekov introduced soon.
Edit: Also the "They've been beamed up... by the gorn" and Pike standing there has the potential to be one of the most horrifying cliffhangers in Star Trek.
While money may not exist on Earth or the core worlds (or the entire federation) per se, there still are Federation credits for use on non-federation worlds, space stations near independent powers, and for commerce with other independent nations (like the negotiations for the Barzan Wormhole). So maybe like on Ferenginar (if it can convert into Latinum) Qo'noS (if they have money), Romulus, or any other economy. So while he may not of used money (as he only bought from Federation worlds), there still is money. It also can be used for worlds which have not made the transition to a post-scarcity/money environment. And to add to that, the Bank of Bolarus implies that some money is used by the federation. So while you are right in that citizens of the federation probably get stuff for free, that doesn't mean money doesn't necessarily exist (which you did point out.)
While money may not exist on Earth or the core worlds (or the entire federation) per se, there still are Federation credits for use on non-federation worlds. So maybe like on Ferenginar (if it can convert into Latinum) Qo'noS (if they have money), Romulus, or any other economy. So while he may not of used money (as he only bought from Federation worlds), there still is money. Also the existence of the bank of Bolarus seems to imply there is a currency for the federation (credits).
After viewing the season trailer and the sneak peek, I became worried about Ortegas as well. Fingers crossed she doesn't have anything happen to her, there is still a lot to develop for her.
Honestly this is probably how it would go: "Hi, my laptop won't turn on. I've tried replacing the power cells, charging it, but nothing's happening. Can you help?" "Hmm... have you turned it off and back on again?" "Umm... it's not turning on." "It's turning on?" "No, it's-" "Hooray. I am smart. I am strong. I will go." "Wait, don't hang-" (hangs up)