Just to clarify, all the adjustment for a truss rod is accomplished by the little nut you removed. Twisting the rod itself won't give you what you want. It doesn't work that way.
Do you still not get enough relief with that nut left loose?
Im not familiar with that one, but if its the "metal protector" that comes up googling the name, certainly not. You need an electronics contact cleaner with a lubricant like deoxit fader lube.
They strings only need to sit in the slot about half way. They look ok from the pic. Don't want to go too low or you'll have more problems.
Check the nut action. Hold the strings down just in front of the second fret. The strings should just clear the first fret.