I've never used a straightedge to check a neck. If you look down the neck length way, you can easily see if there's a twist and see the neck relief (some call it the bow in the neck).
And the thing about neck releif is, it's mainly up to personal preference. Some like a little bit of an arc, some like the neck to be absolutely flat. With me, I prefer my shreddy guitars flat, or close to flat, but my other guitars have some arc in the neck relief.
Another trick for checking a neck is to hold down the high E and low E (one at a time) at the 5th fret and the fret nearest to where the neck meets the body. This will reveal how flat a neck is. You have 6 straightedges built in.
As far as getting the strings lower. You generally do want to keep the bridge level. And you can also swap out the nut for one slightly taller, or shim your existing one with some paper. This will let you lower the bridge a little more before the strings buzz on the lower frets. This is also where neck relief can come into play. It's all simple geometry.