All these questions are answered on the website SpeakoutPetitions.com, but here goes
Can we display a list of signers with this plugin?
Of course, there is a shortcode to do that. Additionally, the list can be formatted either natively or using CSS
Ideally, with first name and first letter of last name, city and state, to keep them anonymous.
There is even an option for people to choose to be completely publicly anonymous. However the admins will see details and they will be included if the petition is sent to a target e.g. your mayor.
Or, can there be an option for the signer to approve showing their name?
I think that is covered by the answers above.
I hope that answers it all for you. Feel free to contact me directly via the contact from on the website.