Those where two separate issues. The apple problems had nothing to do with the incorrect palette.
Uploaded a JASC-file here: goto.glocalnet.net/richard_fhager/db16.pal
Mmmm not i used the palette on your first screenshot with the 4 game mockups. But is possible that to reduce to 16 colors PSP change a little the hues? mmm
I hope to update the apples later
Edit: Yes I see that are some differences but too small. The same apple with the corrects colors are the same to the eyes, particulary i can see a real difference or at least in my monitor.
Edit2 : Updated with the correct colors (I believe)
Noticed that your colors aren't quite correct...have you tweaked them deliberately?
It's not working using the dark brown/voilett as AA on the Apple-outline...it's too close to the black, they just form one thick staircase-line. Use the dark-grey instead. Also, apples are very shiny, they don't get much darker towards the edges...they might even have reflective shine there.
Mm, yeah...that's a big improvement! :)