Q: Mail runs but there is no display.

What happens when you select "New Viewer Window" from the file menu?

Does this behavior persist after...

quitting/restarting Mail?

restarting the computer?

logging in as a different user (e.g., Guest)?

starting up in Safe Mode?

What happens when you quit mail, turn off wifi (or internet), and restart mail (with wifi/internet off)?

Do you recall installing any Mail extensions or plugins?

If this were happening to me, I'd try the following.

Quit Mail.

In Finder, press option and select Library from the Go menu.

Navigate to the Mail folder, then the V2 folder.  Move the MailData folder somewhere else, e.g., Desktop.

Start Mail and test.

Another question:  If you control click on the Mail icon in the dock (or click and hold) you will see a little menu appear.  Here's mine:

What do you have at the very top (where mine says "Inbox - USU ...)?

If you get the "Options, Show Recents, Open" menu that means Mail is not running.  You need to try this after you've launched Mail and the menu bar says "Mail".

Ok. So what is happening is that Mail is getting hung up before it finishes launching.