Do you punish people for sending spam?
Only by making them pay for every mail they send.
Do you filter spam, if so, how.
On the receiving end. A plugin system would let people choose to subscribe to updated blocklists and filtering rules, just like modern adblocking.
Do people need public terminals to access the service?
Same way it is now. The vast majority of people have their own smart devices, and for the ones who don't there's the public library.