We use SendGrid and Mailgun right now, and both of these expose the suppression list, email address, time, and reason code + description. In Sendgrid you can filter, and mass select to remove suppressions easily (which was great). In mailgun I had to export a CSV and just removed them manually as there was not too many across my accounts.

Customers generally cannot change this on their end as far as I can imagine -- this is on the ESP end and is a protection built in because you are sending from their IP / Server and they don't take kindly to that.