On easy mode the levels are not unlocked. If you select easy mode and wait a bit in the difficulty selection menu, you get a scroller that tells you what each difficulty level does (and doesn't).
On normal difficulty the progress is saved, but mind you that it is still not the case if you are using a REU emulation in VICE, because the REU deactivates the saving capabilities of the Easyflah device.
Noticed the screen flashing issue on that configuration and I managed to reproduce it (key buffering glitch). It'll be fixed in the upcoming build. Thanks a lot!