I measure string height at the 12th fret with a ruler that sits flat across the frets (measure to the bottom of the strings). You can also use feelers. I have a specially marked ruler for it, but in my opinion, it’s overkill for a mandolin. When you get as low as a mandolin, I have a hard time reading that accurately. Basically a mandolin should have very low action (compared to a longer scale instrument like a guitar). I just looked at mine and it’s about .050" on the G string (about 3/64"). I could stand to go up a smidge on that on the bass side. As goofy as it sounds, with an adjustable bridge just go as low as you can until it gets a bit buzzy, then go back up enough to clean up the tone. BTW, going to lighter gauge strings will lower the action and heavier gauge will raise the action, so if you are changing string gauges, you might wait until you settle on a gauge you like before you dial the action in real fine.
Assuming there is nothing else wrong with the mando, I would think that the bridge could be shaved to further lower the action. Barring on a mandolin is not the easiest thing in the world, so it doesn’t mean your mando needs adjusting. With that said, if the action can go lower without buzzing, it would help with barres. As far as neck width, there is quite a bit of variety available. Measured at the nut, I have seen them from around an inch to up to 1 1/4" I have one at 1 1/4" and another at 1 3/16", both are considered “wide” nuts.