Why does the drain snake fail to advance shortly after inserting into the drain

It may have hit a pipe joint or bend pipe. Try to slightly bend the microhook head before inserting, and then insert with pressure to help pass through the bend pipe smoothly. If that doesn't work, attempt to reinsert it from a different angle.

How should I clean the micro-hook head after the drain snake pulls out some hair and dirt?

You can try cleaning the microhook head with gloves. But it needs some patience to clean completely. Besides, the drain snake is produced as a disposable product. It is generally recommended to throw it away after use and replace it with a new one.