Re : I can't run processing

Could not run the sketch (Target VM failed to initialize). // What's Target VM???

VM means Virtual Machine. It refers to the Java's runtime itself.

Can you run Processing 1.5.1 and compile programs,
or you simple can't run programs you already had compiled previously?

Sorry I can't be much help. I had no problems at all in running Processing 1.5.1.
Be it Java 6 or 7, 32 or 64 bits!      I believe Processing can even run on Java 4!  

My advise is to delete the subfolder which comes w/ Processing 1.5.1.
And especially, uninstall and then install Java SE JDK of your preference.