Hi Elly,
1. "Do not email" form field is associated to "Do not allow Bulk Emails" field of Contact entity,
"Do not allow Bulk Emails" field is a Two Options(boolean) type field, the two options are Allow(0, false) and Do Not Allow(1, true).
If it's Allow, then "Do not email" checkbox will be unchecked.
If it's Do Not Allow, then "Do not email" checkbox will be checked.
These definitions are managed properties which couldn't be changed.
So actually "do not email" and "allow email" behaviors are integrated into same checkbox.
You can check details in Fields definition of Contact entity.
2. We could run custom javascript on subscription center to check whether "Do not email" is checked when customers submit marketing form(It is "afterFormSubmit" event) with help of Marketing form API.
Below is my demo, please take it as reference:
"Do not email" is the fifth input element in my form page, so I check whether it is checked by document.getElementsByTagName("input")[4.checked function.(HTML element starts index is 0.so we need to minus 1)

I integrated my subscription center with Bootstrap 4, so my design is:
When customer clicked SUMMIT button,
if "Do not email" is checked(document.getElementsByTagName("input")[4.checked equals to true) , show a modal to ask him/her why he/she leaves us,
customer can close modal directly by "Not interested" button or X button ,or click "Ok" to open new page in a new tab and give reason for leaving.

You could run my sample in your portal or website, I have removed my embbed form script,
just add your script inside div which has "content" id attribute to test.(also remember to change index number to get "Do not email" element based on your form layout)
subscription center
Could you let us know why you leave us? Thanks.
In addition, if you would like to redirect customers to new page in current tab instead of opening a new tab, change
window.open(url, '_blank') to location.href(url)
Regards,
Clofly