Nah, there's presumably a third group of force users led by Ahsoka and Ezra (Clone Wars / Rebels characters) that somehow avoid the conflicts that happen in the sequel trilogy. Baby Yoda could be handed off to them.
Turns out Yoda wasn't good at the no relationship aspect of the Jedi Order...
The universe is big ;)
Before that. Baby Yoda's stated age indicates he was born the same year as Anakin Skywalker.
Depending on how they age (baby Yoda is a baby but he's supposed to be 50 years old) there's a decent chance he'd still be a child by the time.
The sequel trilogy started in 2015, The Mandalorian in 2019 - judging from the way the ST turned out, I think it's clear that planning isn't really Lucasfilm's strong suit.
He does so much
Mostly, and yeah, the animatronics for him are excellent.