What are some tips on photographing in dark, mirrored rooms? Should I use a tripod? Often, museums don't allow tripods. If that's the case when I visit, what should I do?
One of those photos is taken with an iPhone 7, another one with a dSLR using iso 1600. I'd say go with the biggest sensor you have, and crank up the ISO to get a good handholding shutter speed, and make sure you do not underexpose. Go handheld. Practice your handholding technique. Go research techniques that use a string or the camera strap to increase stability. Learn about different camera holds.
Figure out how to focus accurately in low light.
And since this is a mirrored exhibit, consider what you're wearing. :)
When shooting in a well-known museum, you can typically go to the museum's website and find their photography policy before heading out there with a tripod strapped to your bag. Easy enough to find out if you can use one or not ahead of time. The Smithsonian website visitor tips page says (bold mine):
Photography
The Smithsonian permits still and video photography for noncommercial use only in its museums and exhibitions, unless otherwise posted.
For the safety of our visitors and collections, the Smithsonian prohibits the use of tripods, monopods, or selfie sticks in our museums and gardens. This is a preventive measure to protect visitors and objects, especially during crowded conditions. We encourage museum visitors to take selfies and share their experiences—and leave the selfie sticks in their bags.
Working members of the media who need to use a tripod or monopod must obtain permission from the museum's Public Affairs Office and must be escorted by a museum staff member while in the building or garden.