Yes. When you've activated a Flipd session, you can still receive and answer incoming calls, and you will also continue receiving text messages as they come in. It's up to you whether or not to exit Flipd and respond to incoming alerts, so be mindful of what you actually need to respond to and what could wait.
If you're using the Focus Lock, only your phone's default apps are accessible, such as Messages, Phone, Calculator, and a few others. Learn more about this feature here.