So say you have a 1080p 60hz monitor, new i7 CPU and a GTX 970 and you're playing a 15 year old game, like UT 99

and it shows you're getting 120 FPS.
1.So since I have a 60hz monitor does that mean I am visually not seeing more than 60fps?
2.If I turn on V-sync it would make the game go no higher than 60 fps, right? if so why would I want to do this?
3. Will I experience any stutter or lag or whatever you call it since my monitor is 60hz but my GPU is producing 120 fps?
4. Now how would things change if I had a 144hz monitor? Would I still turn on V-sync?
5. Lastly when they say "ghosting", which I know what it looks like, what contributes to this? is it only the ms spec of the monitor and that's all? If you have a 1ms monitor you will NEVER see ghosting, true or false?
Its best if you understand how monitors work. They typically refresh 1 horizontal line at a time starting at the top of the screen and working down, this process takes a finite amount of time.
If during a screen refresh the video card presents the next rendered frame before the monitor has finished refreshing the screen, the monitor will simply continue through the current refresh using the contents of the new frame. This means you get a composite refresh that is made up of multiple rendered frames that are stitched together horizontally.
If there is motion and things are changing in game then the content of one frame to the next will be different, and so when you stitch 2 frames together of different content you can see what people refer to as a tear line across the screen, or simply "tearing".
The job of V-sync is to sync up the output of frames with the refresh cycle of the monitor so each refresh of the monitor only displays one full frame, it does this by delaying a rendered frame until the monitor is about to start its next refresh, tearing is then eliminated and your frame rate is locked to 60hz (or whatever your monitors refresh rate is).
There are several drawbacks of V-sync though, if your video card cannot produce a new frame for every refresh then the prior frame is used for 2 refreshes in a row, this means a video card capable of say 50fps that cannot sync at the full 60hz will end up falling back to an average of 30fps, producing frames every other refresh (skipping one), and it will keep falling back slower and slower if your video card struggles to keep up.
The other drawback is "input lag", this is commonly used but really a misnomer, it's not lag on the input but the output, V-sync works by delaying frames and so there's always some level of lag of the frames behind the input making things like aiming with a mouse very hard. Triple buffering can help lower this latency but it's never gone completely.
All monitors of all refresh rates will get tear lines, higher refresh rates primarily give you smoother motion as long as your video card can kick out that many fps. With regards to tearing you get more tearing when your frame rate is higher, the more frames you generate per second the more tear lines you get between each frame, to make use of high refresh monitors like 144hz you ideally want at least 144fps otherwise it's basically a waste, and with 144fps you're going to see a lot more tearing than if you're at 60hz.
Some of the newer monitors come with a piece of technology called FreeSync or the Nvidia propietary version called Gsync which are technologies to alter the refresh rate of the monitor to match the frame rate, it's sort of like Vsync in hardware which eliminates the input lag side of things.
Ghosting is simply related to pixel response time, this is the time it takes in milliseconds (ms) for the pixels to change colour, if you have constant motion in a game then the pixels are having to constantly change colour and you get a ghost image left behind on the screen. Ghosting always exists but becomes very hard to see at the 4ms rates and is perceivably gone by about 2ms or 1ms displays.