If a child was Lord of their castle could or would they be fostered out? Roland I Arryn was fostered with an “Andal king in the riverlands”, granted he wasn’t the lord of the Eyrie at the time, but he was an heir? Also didn’t Jon Arryn or Robert Baratheon want Sweet Robin to be fostered? Would that have continued even after Little Robert Arryn became Lord of the Vale?

Sure. Ned’s older brother Brandon was fostered at Barrowton with the Dustins, and he was the heir to Winterfell at the time. 

And yes, Jon Arryn wanted to foster Robin at Dragonstone to make sure that Cersei couldn’t knock the Vale out of the coming conflict just by taking out Jon Arryn, whereas Cersei wanted Robin to be fostered at Casterly Rock so he could be used as a hostage to do just that. 

As for how long it might have continued, Robert Baratheon himself was both Lord Paramount of the Stormlands and Jon Arryn’s foster son between 278 AC and 282 AC, although he probably would have been finishing up his time at the Vale given that he was about to turn 21.