If a player doesn't remain active on a team's 26-man roster for at least 90 days, their Rule 5 status rolls into the next year, and the same rules apply - they can rehab on the IL, but must be on the active 26-man for 90 days of the 2026 season. I don't know if this rolls over for years and years, though, or if a team would even entertain more than two seasons of the restriction.
I don't know about being "eligible" for the next season. If they remain on the selecting team's 40-man roster for the next off-season, they're protected from the next year's Rule 5 draft. During the off-season, there is no 26-man active roster for a player to be on. Everything is about the 40-man.
So, the Rays selected Lavender. He sat on the 60-day IL all season. At the end of the World Series (I think), the 60-day IL disappears, and players are put back on their team's 40-man roster. The White Sox will have this crunch with Drew Thorpe, Ky Bush and Prelander Berroa. If the Rays keep Lavender, because he didn't spend 90 days on the Rays' 26-man roster, they will have the same restrictions for him. He needs to be on their 2026 26-man roster at least 90 days. They can't option him to the minors without putting him through waivers, then if he clears, offering him back to his original team for $50k (i think). If the original team refuses, Rule 5 restrictions go away, and he belongs to the Rays.