Bae
Your roster is very dependant on the Fleet you end up on ( 777 vs 747) and also your seniority, you really need to see a Junior guys/girls roster.
FWIW the junior pilots on the 777 do almost entirely short ( maybe 6-7 X 3day ) trips, to the Gulf or Eastern Seaboard of the US, with perhaps two days off between ( or back to backing trips if they are struggling to make the monthly target known as CAP). The Senior guys may do slightly better but they as yet don't have a lot of long trips on the Fleet.
On the 747 there are more longer trips to spread around, but the junior guys will still probably be doing 1X5 day trip and 3/4/5 of the shorter ones......
Re: commuting from the European mainland; entirely possible, not always easy, but lots do it.