Navy of conquered countries

Yes indeed, and it's a big ocean. Use subs and DD's to find them. Good luck.

Great indeed.. They should sink or after taking whole country being shown to everyone that country was in war with.

They will slowly deteriorate if you can't find them, but it takes forever it seems. Lack of oil and food and government hurts them but not near fast enough. They will eventually disappear. You could try to change your status with them in diplomacy to peace, helping with war weariness problem and then look for them at your leisure.