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Perfectly fine to adjust the truss rod with the strings on. You normally need to slacken off the inner two strings so that you can turn the hex key in its socket, but no need to remove them. You certainly won't damage it. But if the truss rod nut suddenly becomes very hard to turn, don't force it as you can certainly damage the thread or even break the truss rod. Do not use a lever or pliers etc. to try and get more turning force. If the nut won't move, it's reached the end of its adjustment or there's something in the threads that needs removing.
I find it best to get the neck level before stringing up. String tension once the strings are on and tuned up will give the neck then some concave relief. Normally a bit too much (or more than is necessary), so you may need to then tighten the truss rod slightly, but I always wait until the next day to make any adjustments and give the neck time to settle. If you keep removing all the string tension to adjust the truss rod, the neck simply can't settle down properly and you have to do the final adjustments with most of the string tension applied.
But never do more than 1/4 turn at a time, even if there's a lot of relief to remove. Always check what has happened to the neck are after turning the rod, and if you are close to where you want to be, it's always best to wait a while between subsequent turns of the truss rod to allow the neck to settle. The closer you are, then the smaller the amount you move the truss rod. Once you think you are there, then check again the next day, in case the neck has moved overnight.
As Brendan says, action is really down to the player's preference. I'm a low action person and I play fairly gently. But some players really like to pluck hard at the strings, which gives them a lot of amplitude. The bigger the amplitude, the more neck relief you need to stop the strings catching the frets, and the higher off the fretboard the strings need to be.
The amount the strings vibrate is very exaggerated in that diagram, but you can see that the bigger the amplitude of vibration, the more the neck needs to curve for the middle of the vibrating part of the string to not hit the frets.
What whatever action you decide on, what you do want are the strings to be as low as possible at the nut without them buzzing on the first fret. The higher they are at the nut, the more tension you need to push them down on the first frets, and the more out of tune they will be. But don't adjust the fret slot height until you've got the rest of the action (neck curve and then saddle height) sorted.