Does the 5E open beta corebook or any of the official fiction tell us:

• Which Hantei we've got?
• What year it is?
• How old the Hantei is, and how long on the throne?

I'm hoping I've just overlooked it. Thanks!

They've said only "before the Scorpion Clan Coup" - at least from what I've seen.

Said coup happens in 1123 IC

Hantei Jodan becomes Hantei XXXVIII in 1103 IC, at age 25.

(Source: d20 Rokugan, p. 190)

This means, most likely, they're setting between 1113 and 1123 IC, or years 10 and 21 of the rule of Hantei XXXVIII...

Works for me! Thanks!

Scorpion Clan fic confirms that it's Hantei XXXVIII.