I think I solved the sprite issue. I looked into it more carefully and now I believe the problem is in the uninitalized sprite color table. It seemed to go away if I added COLORSPRITE(0)=0:COLORSPRITE(1)=0 before redefining the sprites on OpenMSX, but in WebMSX it seemed to need something like FORXX=0to31:COLORSPRITE(XX)=0:NEXTXX. I hope this also works on real hardware.
Indeed, that solves the problem in WebMSX.
I'm going to try it on my real hardware tomorrow. I've updated the online version.
Parn: Thank you very much for the time you invested in this. And also thanks to the rest for thinking with me!
/edit 23:27/
I can confirm it also solves the problem on real hardware!
Still some small bugs to fix:
* Sometimes a catched star gets counted twice as a falling star.
* If the starship gets out of the screen, the location isn't always correct anymore.
/edit 00:17/
OK, I think I solved those bugs 
Time to give the wife some attention... nighty night and thank again!
OK, squashed a for minor bugs and added stars with a sine-wave in the 2nd level after a nice idea Sylvester suggested on Discord
Nice!
Seems to be quite challenging 