What evidence is there for additional dark energy coming into existence when space increases?
As I understand since cosmological constant is a 'constant' - increasing the space must generate additional dark energy that fills that space
That's what people say. But we have no evidence whatsoever of anything wherein conservation of energy does not apply. You can find people saying "energy is not conserved in GR", but others dispute this.
This does sound counter-intuitive to me, but I am sure someone has looked at this and tested this thoroughly before making a claim that energy conservation does not hold.
Don't be sure. There is no actual scientific evidence to support the claim.
I couldn't find any articles on how it was measured and confirmed that cosmological constant is a constant, and not that it changes in expanding universe.
There aren't any. But have a google on fine structure constant variation. If that varies over space and time, IMHO it would be reasonable to expect the cosmological constant to vary over space and time too.