From what I understand from doing some searches here on Meta, chat and messages are unrelated. You can imagine a personal message as a normal topic that’s searchable, albeit ‘private’ and permanent [1]. However, a chat is for more ephemeral discussions as in the site settings you can change the retention period of the chats.
A good place to look at other details is in topics like the one below
It depends what type of community you have whether you’d benefit from only chat, messages or both.
You can do this with CSS.
I’m not sure about your third question
I don’t use chat much myself but I think that under the DMs tab it combines individual chats and group chat notifications - not showing the difference. Someone can correct me here.
unless deleted of course ↩︎