How does it work when a Lord Paramount like Tywin or Jon Arryn is given a long term royal appointment (Hand, Master of Coin, etc). Do they have someone back in the Westerlands or Vale running the show for years on end? Does it harm their relationship with their vassals if they’re not considered ‘present’? Or are they just expected to make trips back home as often as possible?

As often as possible is a bit much, but frequent trips are a part of the deal. Tywin very much commuted between King’s Landing and Casterly Rock (especially since his wife wasn’t welcome at Rhaella’s court after a point) and even “worked at home” for about a year. 

But in the main, there’s two things you need – one, a strong castellan to rule the roost back at home and keep your bannermen in line, and two, lots of patronage both in terms of employment but also as invitations to come hang out at court, so that you can bring your bannermen to you. 

And that’s the kind of push-pull thing about long-term jobs. Yes, it means you’re away from home, but it also means you have lots of royal goodies to hand out, which means your bannermen have more reasons to be on your good side.