They can improve. A palette-limited source is restricted to 256 colours, and video codecs are full-colour, so banding in the source stays but no new banding is introduced. Where MOV ni QuickTime, Apple-native, àti ile tí a lò fún ProRes àti àwọn ààyè-iṣẹ́ àwọn ààyè-iṣẹ́ alpha tí MP4 kò lè gbe, the conversion generally makes gradients look better, not worse.
No — the source has no sound, so the MOV is silent. Some platforms prefer a video to have an audio track; add a silent or music track afterwards if a specific platform complains.