This is an interesting one and I'll start off by saying that I do not have the answers
That said, I do know that when you are playing OOTP with historical minors, teams can go on hiatus for multiple years and still pretty much exist. By that I mean that (in this totally made up example) if the Anytown Hooters play in the Good Ol' League in 1925, don't play in 1926 & 1927, and return - to any league - in 1908, first, you would be able to find the team (essentially it's prior year history) via a search or thru the historical standings of the league. And then, as long as the Historical Minor Team ID is the same for the 1928 team as it was for the 1925 team, then there will connectivity in the team's history. But that's for OOTP with historical minors. Since you are using an 1897 example (which is before OOTP introduces historical minors), I'm assuming you have a fictional setup.
At any rate, the answer to your questions might depend upon how you pause those teams...
How it would work in the OOTP
historical world is that if a milb team disappears - either temporarily or permanently - this would occur when the game transitions from the postseason to the offseason (so typically mid or late October). And if the team is affiliated with a major league team, then those players would be distributed elsewhere into the MLB team's system, or - if the disappearing team was that MLB's team's only affiliate - onto the MLB team's reserve roster. In either case, if there wasn't room for all of the disappearing team's players in the MLB teams' system, then some of the players would become free agents... If, however, the disappearing team is not affiliated with an MLB team, then their players become free agents. I don't think there's any scenario where players remain on a defunct team...
So ya, all of that above is probably not all that helpful since it's historical-based

What I suggest is to make a copy of your league and run a test or two to see what happens. And then definitely report back here

Good luck!