Can a professor do an internship?

Yes, but they are usually called "visiting researchers" or, in some cases, "residents", rather than "interns". See for example Facebook, Microsoft, or Google.

Yes, absolutely. I am not able to answer for every discipline and country, but this is common in many places. There are company schemes, government schemes, charitable schemes and university funded schemes to support such arrangements.

For a specific example, in the UK the Royal Academy of Engineering provides Industrial Fellowships to support such activities.

Mathematics professor Robert Talbert, in a 2018 article titled "What I Learned On My 'Secret Sabbatical' As a Scholar-in-Residence at a Private Company", described how he was an intern for a year at the furniture company Steelcase:

Last fall I started my first day on the job as an embedded faculty member with a corporation—as a scholar-in-residence at Steelcase Education. But don’t be too impressed by the title; according to the employee system, I was just an intern.

Actually intern is probably the best lens through which to look at what I’ve been doing at Steelcase for the last eight months. I was nobody special: Just a guy, at the bottom of the org chart, working with and around a lot of people smarter and more talented than I am in any number of ways, and tasked with making their work and the collective work of the organization better. It’s kept me a little bit humbler than I would have been otherwise.