Is there a function that can render the text to HTML including Paragraphs and Bolds etc?

I have changed the notification function to send everything in HTML but the text of the entry does not have < P > tags.

Are you talking about the notification email sent by MT? This will include the HTML tags from your entry but not parse them. The p tags are inserted if you have your text formatting set to "convert breaks".

Yes. What I want to do is modify send_notify function at CMS.pm to parse the entry content, same way as MT do it to generate the regular page.

I believe the Notification needs to be improved. First to allow HTML email and second to templatize the email.

I believe this is on the "requested features" list; it's something that I've wanted for a really long time too. In my opinion, the notification feature is one of the weakest parts of MT. Hopefully they will be coming out with an improved version in MT 3.0.

A temporary solution on notifications that you might look into is Bloglet. I have it pull in my RSS feed and it sends out an email once a day with the new posts for that day, all HTML beautifully rendered.