Can the small wood particles that fly around when you cut wood get in your lungs and damage them?

That depends on the amount. Your lungs have a transport system where mucus traps fine particles and moves them from deep in the lungs to the outlet where they can be coughed out or swallowed. This is active for smoke or fragrances and particles. That system is occasionally overwhelmed like we see for black lung disease in coal miners. Where a N95 mask when possible.

Agree with Dr. Ferguson. Wear a mask when working with wood. Many tools that generate saw dust have vacuum attachments. You could keep "shopvac" hose near the working end of the tool to remove saw dust. Wish you good health!