Pages are just static HTML. I frequently load them into codepen and debug there so I don’t need to make round trips during debugging. You could also do that locally.
No, but there are orther third party integrations available with actual search products. Take a look in the University.
As for the publishing issue; If it was pervasive there would be a notice on status.webflow.com which shows as clear. Try publishing again and be sure to follow the link from your project settings or from the designer. I this persists, then open a ticket with support.