Actually 1920 x 1080p @60Hz has a higher pixel clock than that.
There is a back porch then the 1920 pixels then a front porch and Hsync on the horizontal line. VESA standard makes the horizontal line to be equivalent to 2200 pixels across.
Similarly on the vertical, you have a vertical back porch then the 1080 lines then a vertical front porch and then the Vsync, making it 1125 lines worth of signal.
So the pixel clock is 2200 x 1125 x 60 (for the number of frames per second) equal to 148.500000 MHz and each pixel consisting of red, green and blue (sent in parallel via LVDS along with the clock sent via LVDS as well to the LCD.
Edit: Essentially pushing a bit under 425 MB/s (where MB is 1024x1024 bytes) worth of data (or 3.6Gbps if you think of it as network speed where Gbps is 1000x1000x1000 bits per second)
1) when my monitor says "60 Hz" does this mean that at best there will be 60 frames painted on the screen per second regardless if the graphics card displays for example 100 FPS?
Yeah, you won't see the difference, but for games (1st person shooters) an FPS is a game clock tick, so if the game updates 100 times per second your gameplay and reaction to others has more resolution, meaning you can shoot faster or react faster the more FPS the game runs at.
2) What do I gain by buying a monitor that claims 144Hz?
That will let you run 3D games (as in stereoscopic) at 60Hz since it has to draw double the frames plus in between eye blanking. Then again, that will half the game play so you are back to the response time of the game running at 60 Hz.
Some TVs can even do 400Hz and it seems it helps the color rendition but I'm not quite sold on that since I haven't seen one in person. I guess they can interpolate frames to give a smother and more fluid look.
Professional gamers do benefit from higher frame rates, and they develop good timing muscle memory that allows them to do repetitive sequences that gives them an edge, but that is after playing a lot, I mean like any NBA professional played B-Ball on every single waking moment. For your average gamer there is no difference.