Hi @o_x,
- Can campaigns run without setting expiration so they run forever? - Yes.
- What happens if a campaign was set an expiry date, how do we resume/remove the expiry date? - You can simply remove the start and end date on the properties of the campaign.

3. How do I quickly check which campaigns had expired? - You can either setup a query to look at the campaigns that are expired or maybe use the timeline option from ACS.
https://experienceleague.adobe.com/docs/campaign-standard/using/getting-started/marketing-plans/timeline.html?lang=en
4. Also please kindly share documentation around this.
https://experienceleague.adobe.com/docs/campaign-standard/using/getting-started/marketing-plans/programs-and-campaigns.html?lang=en
Note: When you don't set any start and end date it will show the execution status as In progress and after the end date it will show as finished. There's no expired status.