There isn’t a way to create an incident without assigning it to someone; this is to encourage quick resolution of your incidents. If it is unclear who will own the incident at the time it is created, it is recommended to assign it to an escalation policy where multiple people are notified. This way each of the users is made aware of the incident, and the user who will own it can acknowledge it.
If your concern is that you will wake people up for incidents that can wait until business hours, you can use defined support hours to get around this. Here is a related community post that outlines how to set this up. If your use case requires having incidents unassigned, a possible workaround would be to create a “Dummy user” and assign it to them.
By the Activities section, are you referring to the Activity for the Past 7 Days listed at the bottom of the main incidents page in PagerDuty? If so, this section does include Acknowledge events.