Note that Castle Garenbrig's second ability says "T: Add G". Normally, this ability isn't affected by Damping Sphere since it doesn't produce two or more mana.
If you control Damping Sphere and activate Castle Garenbrig's third ability, Castle Garenbrig will produce C instead of the type or amount of mana it would otherwise produce, since it would otherwise produce six G this way. You can spend this C "only to cast creature spells or activate abilities of creatures".
If you control Damping Sphere and activate Castle Garenbrig's third ability, Castle Garenbrig will produce C instead of the type or amount of mana it would otherwise produce, since it would otherwise produce six G this way. You can spend this C "only to cast creature spells or activate abilities of creatures".