Also slightly faster version of IFA=3THEND=D+VELSEIFA=7THEND=D-V is to write it as
D=D+V*(-(A=3)+(A=7))
Are you sure?
Actually no 
but there must have been a reason why I did it like that in the past when I did use BASIC.
So I tested it again using this program.
10 DEFINT A-Z:D=0:V=3 20 A=7:GOSUB30:A=3:GOSUB30:A=0:GOSUB 30:END 30 PRINT "a=";A;" ";:TIME=0:FORB=0TO1000:IFA=3THEND=D+VELSEIFA=7THEND=D-V 40 NEXT:PRINTTIME;:D=0 50 TIME=0:FORB=0TO1000:D=D+V*((A=7)-(A=3)) 60 NEXT:PRINTTIME;:D=0 70 TIME=0:FORB=0TO1000:D=D+(A=7)-(A=3) 80 NEXT:PRINTTIME 90 RETURN
And the output was
a= 7 332 403 312
a= 3 267 401 312
a= 0 249 378 313
So I conclude that
a) my method is indeed slower but always uses a constant time while the if method varies depending on input
b) the impact of the multiplication is significant, almost 25% extra time needed
Now for the sprite issues.
The thing is that I actually do set the sprites after each page swap.
530 SPRITE$(0)=CHR$(16)+CHR$(56)+CHR$(124)+CHR$(254)+CHR$(124)+CHR$(56)+CHR$(16) 540 SPRITE$(1)=CHR$(129)+CHR$(129)+CHR$(129)+CHR$(195)+CHR$(165)+CHR$(153)+CHR$(255)+CHR$(255) 550 RETURN
Jipe found a workaround, but I wonder if doing a 'screen,0' before the first SPRITE$(0) and setting the colors with COLOR SPRITE$(0)=STRING$(8,7):COLOR SPRITE$(1)=STRING$(8,6) would help?
I wrongly assumed CLS initialized the sprite area, but unfortunately it doesn't. 
So you would have to manually initialize the sprite area, I believe. This code can do it:
FOR I=0 to 127 STEP 4:VPOKEBASE(28)+I,217:NEXT
This would make every sprite invisible by default, and theoretically it wouldn't matter what's on the rest of the SAT.
Indeed... The POKING instead of Set Page didn't work out. CSL did neither and the VPOKES still gives me the same problem. Only creating the sp.bin file Jipe proposed seems to solve it. But if I need to have disk-access anyway, I might consider just loading the stages one by one (and maybe add a loading screen with some information of the upcoming level).
I was hoping that I had a small oversight, but it seems to be some weird MSX Basic behavior.
The way of controlling the spaceship thegeps proposed is indeed a lot quicker and even more stable in time:
So I have updated that. I also found a bug that caused the number of stars remaining to be printed at the star position in the spacequake stages. Anyway, the updated version can be found here: https://test.file-hunter.com/
The sprite issue isn't implemented in this version, as I am going to move to loading the stages on the go.
Thank you all so far for thinking with me :)
My request for music remains though!