Only light can pass through the gem, and nothing else.
The Minimus Containment option of imprisonment says:
Light can pass through the gemstone normally (allowing the target to see out and other creatures to see in), but nothing else can pass through.
I've emphasized the relevant portion - nothing else can pass through. Not even the Paladin's magic. But why?
Abilities that affect other creatures generally require a clear path to the target:
To target something, you must have a clear path to it, so it can't be behind total cover.
If you place an area of effect at a point that you can't see and an obstruction, such as a wall, is between you and that point, the point of origin comes into being on the near side of that obstruction.
The paladin's class features and spells, things that magically affect others, are not light. Since the gemstone only allows light to pass through, the paladin does not have a clear path to the target for these abilities - their abilities cannot pass through the gem that they are inside.
To see this another way, we can emphasize that the target can't be behind total cover. Everything outside the gem has total cover with respect to the paladin, so the paladin cannot target anything outside the gem.