Do the league cup final losers get a European place

[Somniloquism]

If Arsenal win it then it goes to the 6th placed team in the league

Assuming Arsenal finish in the top 5 places. I know it is a given but we can only but dream that the top 4-5 places are not predictable at the start of every season.

[Somniloquism]

And if one of the top 4 win the fa cup then 7th place gets a European spot so plenty to play for yet.

No, Runners up of the FA cup get the European place.