Hey Manager! The Key Attribute increase gained with Player Tiers does not downgrade with the new season. It is permanent and active until the player retires. However, please keep in mind that this doesn't prevent the star loss of the promotion.
As for the second question - all Gems, that are not spent in the meantime, are stashed and carried over into the next season.
For more info about Player Tiers, make sure to check out the article: https://norde.us/ia

Hope this helps! macmag123