As far as the priest's prayers go, that's how it was up until the mid 60s (and priests can still use it now, of course). As far as the penitent confessing in latin, the church has never required latin competency for the laity, so confess in whatever language is most comfortable between you and the confessor. Usually, that's the vernacular of your given country, but in some unique situations, it could be latin if the priest is comfortable, you're a latinist, and there's otherwise a language barrier.
Regarding your assumption: The ritual of private auricular confession as we know it is a product of the early Middle Ages rather than the early days of the church - by which time Latin was being replaced in popular usage by its Romance progeny and other vernaculars. (Of course, there were tales of how Irish clergy, schooled in purer Latin because it wasn't related to their own vernacular, took a dim view of Latin as actually used in Rome in that era.)
Most of our priests give the absolution in Latin, although recently a visiting priest from the Middle East always gave the absolution in Aramaic.
As for language to confess in, the signs on the door alway list the languages that the confessor speaks... Polish, Spanish, French, Aramaic, Arabic... are regularly listed.
Absolution in Latin was very common until the post V2 conciliar absolution formula came about.
Dominus noster Jesus Christus te absolvat; et ego auctoritate ipsius te absolvo ab omni vinculo excommunicationis, sespensionis et interdicti in quantum possum et tu indiges. Deinde, ego te absolvo a peccatis tuis in nomine Patris, et Filii, et Spiritus Sancti. Amen.
I once had a priest (priest-monk) who was quite old and going senile (the full story is kinda longish) hear my confession. When he got to the absolution he paused... said "I can't remember the English, so I'll use Latin"... pause... "I can't remember" I said "Ego ..." which I knew (and it was all I knew) began the essential part, he said "Oh yeah" and hammered it off.
Concerning examination of conscience tools in Latin: I own a prayer book containing several (mid 20th century, comprehensive handbook of Latin prayers intended for private use by seminarians)
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