For most skill BGS: Probably. For the very first game we still got BGS, but staff picked the research/experimentation subjects. Since the ability to enter a BGS on the website is linked to if you are registered for that upcoming event or not, I would guess that as long as you can enter the new character into the database before the BGS deadline then you could still enter BGS for them as normal.
For lessons: no. Doug never attended any lessons IG, so they wouldn’t be able to come into game 1 with a lesson learned. (And since you need IG info to use your focus to learn a lesson in BGS they probably can’t do that either) And James isn’t attending the event the lesson BGS is being submitted for, so he doesn’t get to submit a lesson either I guess. I would wait for the staff response on that last one. Maybe Athena can tell us a bit more how playing and switching between two characters worked out for her.
That’s just my best guess though, I’m not staff of course.
You cannot submit BGS for a new character before they come into play
You cannot submit BGS for a character for a game they are not attending.
So, if you switch characters, you cannot submit any BGS in that cycle. You can submit BGS (including a lesson) for your old character for the game in which you resume playing them, and you may do so for your new character for any subsequent games.
To be entirely clear I’m understanding, a new character’s research will always be undirected, and cannot have any player input before game. Is that correct?