It looks like there is something wrong with the launcher. The launcher will launch even if there is no minecraft folder. I would:
- Delete your old minecraft launcher.
- Download a new launcher from http://minecraft.net/download.
- Make sure to backup your saves!
- Delete your
minecraft
(or .minecraft
on Windows and Linux) folder.
- Open up your newly downloaded launcher.
- Click
Options
.
- Click
Force Update
.
- Now log into Minecraft normally.
- Close Minecraft.
- Copy your saves folder into your
minecraft
(or .minecraft
on Windows and Linux) folder.
If you have any problems, make sure you have the latest version of Java installed. I know 10.5.8 cannot get Java 7, but Minecraft will run on Java 6.
Another possibility is that you have a PowerPC (PPC) processor. Computers with a PPC processor cannot run Java 6, and are stuck on Java 5. Minecraft is discontinuing support for Java 5. The changes are planned for Minecraft 1.6, however.