The most important setting for this is the internal resolution setting, which you can find at Graphics > Enhancements > Internal Resolution. The Enhancements tab also lets you set some other settings that can make games look prettier, like anti-aliasing.
HD texture packs are not needed to run games in higher resolution, but they can be used to improve the graphics quality by making textures more detailed. The reason that they are called "HD" texture packs is that the low-quality textures that are used by default in GC/Wii games often are detailed enough to look fine in SD, but not in higher resolutions. Not all games have texture packs available though, because it takes time and effort for people to remake textures.
I know very little about the Dolphin Bar, but I think the mode where it acts as a mouse is the wrong one. There should be a mode that gives the Wii Remote's input directly to Dolphin instead of having movements translated to mouse movements.
HD texture packs are not needed to run games in higher resolution, but they can be used to improve the graphics quality by making textures more detailed. The reason that they are called "HD" texture packs is that the low-quality textures that are used by default in GC/Wii games often are detailed enough to look fine in SD, but not in higher resolutions. Not all games have texture packs available though, because it takes time and effort for people to remake textures.
I know very little about the Dolphin Bar, but I think the mode where it acts as a mouse is the wrong one. There should be a mode that gives the Wii Remote's input directly to Dolphin instead of having movements translated to mouse movements.