Deleting mails from Apple Mail Yosemite 10.10.5 In my previous iMac which was 8 years old, I had it set up to be able to automatically connect to my usual @outlook.com account whenever I clicked on the mail icon to forward a webpage to someone.

The way it was set-up, it would only be able to do this function and would not receive all my incoming @outlook.com emails. I have followed your instructions and the new Mac is now the same.

But now I want to know if I could delete all the mail that was downloaded into Mail before I disabled the incoming feature.

I hope you are able to follow all that I've typed here. If you understand what I'm trying to do, please instruct me what steps to take to delete all those downloaded email from Mail.  Also, advise me if this is not a good action to take. I just feel that it is sufficient to have my emails in the @outlook.com inbox rather than have a copy of the incoming mails in Mail, too.
Some of it depends on what type of account @outlook.com accounts are but I am quite sure that it would be IMAP. As such, if you delete your messages, it will delete them from your live email account to, which obviously could come with unintended consequences. To avoid that, I would recommend first confirming that messages are only set to download when you click Refresh. As I don't have Outlook I cannot confirm or guide you on how to do that. 

After that is set, I would delete the account from Outlook and then re-configure it. Doing so will remove messages from your computer without deleting them from the cloud. You should then be able to send emails as desired. 

The other thing to consider is that unless you are running low on disk space, there is no need to worry about having some unneeded messages sitting in your Outlook application. If it were me, I would turn sync to manual and leave them there, likely. 

Edit:
It took me awhile to grasp your question, and it looks like I partly failed. It looks like you don't want your messages sitting around in Apple Mail.app. As such, the above would apply. However, OS X does allow you to set default Mail programs and you can likely set Outlook to that function. With that, you could use Outlook as your primary, and only, email client and delete the account from Apple Mail.