The conjunctive structure "arrogance AND shamelessness" should not mask the meaning behind the phrase.

The term بجاحة means pride in the worst sense: arrogance. It is usually used not just in the sense of an exaggerated self-regard, but in the sense of such self-regard being displayed with ugly lordliness. The best choice of word will depend on the context. For example, for the first sentence in your list of examples, I think that the best choice would be "triumphalism."

The word صفاقة is used in the sense of displaying such disregard for propriety as to demonstrate lack of any sense of shame. All the terms suggested by my colleagues are excellent choices: brazenness, audacity, etc.

The point, however, is that the conjunctive form, using "and" works fine in Arabic, but should not be imitated in English. The meaning is "arrogance coupled with shamelessness," in other words, "shameless arrogance."