Email

Please find below an excerpt from a textbook.

Quote

A: I’d recommend talking to your supervisor in advance to determine whether a phone call is necessary or just an email (1) is enough.

B: Some managers prefer what they see as the efficiency of email (2). Others are more traditional and want the personal touch of a phone call. And then there are those who are OK with an email (3), but expect employees who are feeling under the weather to follow up with a phone call.

C: The advantage of phoning in sick is that you and your boss can talk about how long you expect to be away, and what work needs to be done in your absence.

D: And I guess they could also get a sense of whether you’re faking it. Not that any of us would ever do that.

A: Heaven forfend. In my humble opinion, email (4) is just fine. Most employers are in the habit of checking their email (5) before they leave for work in the morning. ……..

Unquote

There are five “email”s used in the above dialogue. I understand the first and third ones are “messages sent by email” while the second one is used to mean “a way of sending messages by means of computers”.

I have the following questions:

1. Can the second one (“email”) be replaced by “an email” or “emails” though the meaning of the word changes?
2. Can the fourth one (“email”) be replaced by “an email” or “emails” though the meaning of the word changes?
3. I have a problem with the fifth one (“their email”). Shouldn’t this be “their emails”, “their incoming emails” or “incoming emails” instead?