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Nous avions un peu de temps avant le départ, alors une autre élève a suggéré de prendre des photos à ¶
"Pdans un « purikura »."

À + a capitalized name afterwards would really make it sound like it's a city or something

C'est quoi P, un purikura ?

Same thing here, it's a kind of object, a common noun, so it's not capitalized and the traditional rules regarding articles apply

C'est difficile à expliquer, mais Pun purikura, c'est un type de cabine photographiquephotomaton, où on peut prendre des photos et les retoucher/modifier avec des couleurs, des illustrations ou quelque chose d'amusant/des choses amusantes.

Such an object is rather called "un photomaton", at least in France (I think I've never heard "cabine photographique" once in my life)

On as'est promis de s'écrire.

"se promettre" (with a "se" indicating a reciprocative action) would be better, since it's a promise you all made to each other

Lorsque l'autobus est arrivé, je suis entrémonté dedans et ils me faisaient signe au revoir sur le quai jusqu'au départ.

In a pinch you could say "Rentré dedans", but it would be awkward, since "monter" is pretty much the standard verb for vehicles
"ils me faisaient au revoir" is enough to express the idea they were waving

Ça serait impossible.

Or => CE serait impossible (a bit more formal/elegant)

Après tout, quatre années après, j'en ai épousé l'une.

"année" is possible after numbers, but it's rare, pretty literary and really emphasizes the amount of time that went by
"L'un" is pretty much always said about subjects, in literary contexts