Mac Mail very very messed up

Question

Since I download the last updates for my Mac, my mail program has become almost unusable.
1. It will not shut down - have to force quit and computer won't shut down until this has been done.
2.  Nothing appears in my 'sent' folder when I send e-mail - even though recipient says they have been received
3.  Mail hangs when being sent - on occasion taking up to 5 minutes to send an e-mail.
Please can someone tell me what is going on here.  I have deleted all my e-mail accounts and set them up again, checked and double checked all settings, run repair disk permissions until blue in the face and nothing seems to help.  I am not a 'techy' person and so can't go into the 'inner workings/code' etc to check things, just don't know how.
This is just particular to my Mac, as when sending mail from my iPad or iPhone it goes straight away and I have no problems at all, therefore nothing to do with my router, isp or anything like that.
Some support on this would be great as I am pulling my hair out. 
Thanks - from a very upset Mac user

Answer 1

Quit Mail. Force quit if necessary.
Back up all data before proceeding.
Triple-click anywhere in the line below on this page to select it:
~/Library/Mail/V2/MailData
Copy the selected text to the Clipboard by pressing the key combination command-C. In the Finder, select
          Go ▹ Go to Folder
from the menu bar. Paste into the box that opens by pressing command-V, then press return.
A folder window will open. Inside it there should be files with names as follows:
          Envelope Index
          ExternalUpdates.storedata
Move those files to the Desktop, leaving the window open. Other files in the folder may have longer names that begin as above. Move those files, if any, to the Trash.
Relaunch Mail. It should prompt you to re-import your messages. You may get a warning that the index is damaged and that Mail has to quit. ClickOK. Typically, the process takes a few minutes, but it may take hours if you have gigantic mailboxes. In that case, you may be able to speed things up by temporarily adding your home folder to the Privacy list in the Spotlight preference pane. Remove it when Mail has finished importing.
Test. If Mail now works as expected, you can delete the files you moved to the Desktop. Otherwise, continue.
Please follow these directions to delete the Mail "sandbox" folders. In OS X 10.9 there are two sandboxes, while in 10.8 there is only one. If you're running a version older than 10.8, this comment isn't applicable.
Triple-click anywhere in the line below on this page to select it:
~/Library/Containers/com.apple.mail
Right-click or control-click the highlighted line and select
          Services ▹ Reveal
from the contextual menu.* A Finder window should open with a folder named "com.apple.mail" selected. If it does, move the selected folder—not just its contents—to the Desktop. Leave the Finder window open for now.
Log out and log back in. Launch Mail and test. If the problem is resolved, you may have to recreate some of your Mail settings. You can then delete the folder you moved and close the Finder window.
This action will delete any custom Mail stationery that you have created. If you want to preserve it, ask for instructions.
If you still have the problem, quit Mail again and put the folder back where it was, overwriting the one that may have been created in its place. Repeat with this line:
~/Library/Containers/com.apple.MailServiceAgent
Caution: If you change any of the contents of the sandbox, but leave the folder itself in place, Mail may crash or not launch at all. Deleting the whole sandbox will cause it to be rebuilt automatically.
*If you don't see the contextual menu item, copy the selected text to the Clipboard by pressing the key combination  command-C. In the Finder, select
          Go ▹ Go to Folder...
from the menu bar and paste into the box that opens by pressing command-V. You won't see what you pasted because a line break is included. Press return.