My recommendations are:
- If you have a floating Live Chat button in the bottom of the page, then a simple “Help” menu item in the header is enough.
- If you don’t have that floating button, then you could add “Live Chat” or “Chat” as another menu item in your secondary header menu (on desktop)
- Add a phone number if calls are an important channel of communication for your store (for your audience)
- An email address or email icon isn’t necessary but if you have the space for it in your header, it’d be something for you to test.
Note that if your demographics are young(ish), then Live Chat is an absolute must.
As for putting an email in the header, as you can see in the examples below, only Crutchfield has that option, but it is in a side panel triggered by clicking a menu item or button in the header. So the email address isn’t visible. I have seen this on some stores I looked at, but it is not very common.