html field

The HTML field is just for front-end display in the form, not for the notification or entry. You can manually add such blocks of text in your notification by removing the {all_fields} and including the fields one at a time in your notification.

In Notifications, your body/message probably just says {all_fields} currently.

Remove this and look for the down facing arrow to the right side of the field. You can add in merge tags for the field values you want to send.

You can construct your message like this for example:

Hi {firstname:1},

We just wanted to send you a copy of the information you submitted to us:
{mergetag:2}
{mergetag:3}

Also, please be sure to visit <a href='http://google.com'>Google</a>. 

All my best,
David

See how I was able to use text and HTML and not just the merge tags to include more information? That's what you can do as well.

The HTML field cannot be added to the notifications. You can add the same content though. So you would copy the information from your HTML field and put them into your e-mail manually.

UPDATE: see the last post in this thread

Not currently, no. You can make the details in the notification conditional as well using Conditional Shortcodes - http://www.gravityhelp.com/documentation/page/Shortcodes

Unfortunately, you can only use one object as a comparison in Notification messages unless you use full Conditional Notifications, which would require a great deal of work to create and manage.