The resolution would have an "i" after it - 1080i is interlaced, 1080p is progressive. Your monitor's manual should say something about that, but almost everything is progressive.
A 60Hz monitor updates every 1000ms/60 ~ 16ms. If your GPU were to render a frame 1ms before the refresh, then run at 120fps it would render another 8ms later, or at t=7ms, and another at t=15ms. Your monitor would show frame 1 then frame 3 -- completely skipping frame 2 because two complete frames were rendered between a screen refresh. In general, if N complete frames are rendered between a refresh then N-1 are skipped. (Tearing is a separate issue, it's possible for 2 of the frames to be poorly merged - ie, the top 20% of frame N-1 and the bottom 80% of frame N. If a big change happened between those two frames - ignoring the other N-2 entirely - then you'll see that as a tear.)
FreeSync and G-Sync exist to make the screen refreshes "random" - it could be 60Hz, but they're not every 16ms. Instead, they're "kicked" off by the GPU rendering a complete frame. So if it took 10ms, 15ms, then 30ms to render a frame the first frame is up for 10ms, the second for 15ms then the third for 30ms. On a normal monitor, you get a refresh every 16ms regardless of what the GPU is doing.
Now, the downside to increasing the settings until you hit 60FPS. First, do you mean 60FPS max or 60FPS avg? If maximum, then you're spending a lot of your time with sub-60FPS. So let's assume 60FPS avg. Here you'll also encounter areas of the game where it dips to 45FPS, 30FPS, etc. In these case you may have the same frame displayed for multiple refreshes - meaning you don't get a new frame every 16ms, but perhaps every 32ms, or every 48 ms. This creates stutter and lag. If you keep the settings up such that you are getting 100FPS on a 60Hz monitor, then you at least are getting a new frame every 16ms and you shouldn't see any stutter. You may see tearing, however, if the frame changes while the monitor is drawing it.
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