OK, I see my last post was a bit late. But you've made progress, good.
1) The white thingy is called the nut.
2) Raising the bridge saddle does NOT make the string tighter(well, temporarily it does, but after raising it you tune the string back down to correct pitch. Once you do that, it's the same tension as before).
3) The neck should look almost perfectly straight. If it is not, the truss rod needs adjusting. I don't think you have the proper experience to do this yourself.
Now, if the string is rattling at the end where the nut is, the string slot is cut too deeply, or at the wrong angle, or it's too large for the string. The string needs to fit snugly in the slot. One way to temporarily fix this is to place something in the slot, under the string.
A better way is to remove the nut from the neck, and place something under it, raising the whole nut. Then re-adjust the depths of each slot to get the correct string-to-neck distance. Again, you need knowledge and certain tools to do this right.
Your best bet at this point, I think, is to get help from someone you know that knows how to do this stuff, or you might end up worse off than before. But good for you for making progress.