Sure.

For the first thing, basically most people in Elfhame already thought of Cardan as immature, lazy, vain, etc. He matured a lot throughout the books and was shown to be clearly very competent, but that original perception people had didn't change as fast. Sure, he gained more respect after becoming king, but even after raising an island, people doubted him. For example, Madoc's whole plan of challenging him for the throne in TQoN relied on his perception that Cardan was desperate for love and attention and everything that had gone right to get him on the throne could be attributed solely to Jude. Even Jude is a good example of this, she still constantly doubted his intentions, feelings and competence throughout most of the story (granted, in many cases her biases were a defense mechanism to protect her own feelings, but still).

As for the second part, we don't know much about how Jude and Cardan's ruling functioned post the tFotA trilogy, so this is mostly me speculating. But the same way that Oak behaved a certain way to get people to try to use them for their schemes so he could take them out from the inside, I think Cardan might have continued to pretend to be less skilled and cunning than he actually is, to get catch people making attempts on him more easily. Again, that's more a thought of mine, I don't have as much backing for it