No. There is no such provision
No, but the champions will automaticaly qualify for the next year's Europa League.
No but they do qualify for the Europa League.
They didn't qualify for the UEFA Champions League but they did qualify for the Europa League.
No. They qualify for the europa league next year no matter what league position.
Yes, you do.
Only if they qualify through their own domestic league.
No, they go into the group stage of the following season's Europa League.
Yes. In which case the club with the highest league position that hasn't qualifed for Europe by other means(cup competetions etc) will usually get a place in the next seasons Europa League, usually entering at the 3rd qualifing round.
No. You qualify for the next season's Europa League, unless you qualify for the Champions League in another way (e.g. winning your domestic league).
The top 4 all qualify for the Champions League, with the fifth team qualifying for the Europa League. The FA Cup winners and League Cup winners also qualify for the Europa league.
The FA Cup winners qualify for the following seasons UEFA Europa League. This European place applies even if the team is relegated or is not in the English top flight. However, if the FA Cup winning team has also qualified for the following seasons UEFA Champions League, then the losing FA Cup finalist is given the Europa League place instead. FA Cup winners enter the Europa League at the Group Stage. Losing finalists, if they enter the Europa League, must begin earlier, at the play-off or Q3 stage.The FA Cup winner also qualify for a single match of FA Community Shield against the Premier League Winner a week before the start of next season Premier League.The winner of the competition also earns around £2million in prize money.
This is not possible as the winners of the Champions League and Europa League get direct entry to the next UEFA Champions League. Hence this case can't happen in the Champions League.