Well, I never really was the master of nets.

I do (and really should've) know, though, that it was wrong of me to rotate the texture verically like that. It's supposed to be upside down, so that the game will rotate the texture itself. If you rotate the texture for the game, it will only be rotated again. Probably one of Nintendo's emulation paranoia things; I don't know, they do this with many other games.
Your graphics card should not affect how textures are wrapped, I don't think.
Rice's I'm not sure about. It may fix the black background of the tree texture, which should really be transparent, with some alpha settings. Try checking "Normal Alpha Blender" in Jabo's advanced options.
To be honest, I'm feeling too lazy to actually get into the game and dump the texture myself, look at what's going on and why just the native texture has no issues, maybe others will like that job more than I will, but I'll try to answer your questions.
But now I think I see the point of Xenobond's question. It's the program that's saving it. The native texture had a different formatting level of PNG that affected the background's transparency storage method, ..I guess.

Try saving changes to it with various programs; I used ArtIcons Pro.