How come that an LCD with 1366 x 768 native pixel resolution can support 1080i/p which is 1920 x 1080?
An LCD TV with a native resolution of 1366 x 768 can only display video signals with a resolution of exactly 1366 x 768.
An incoming standard DVD signal has a resolution of 720 x 480 (NTSC). A fixed pixel display such as an LCD TV must first scale the image to the panel's native resolution (1366 x 768, for example) in order to be able to display the image. The job of scaling the image to the panel's native resolution is done by the TV's internal scaler software.
A standard DVD signal upconverted to 720p by an upconverting DVD player still cannot be displayed by the 768p LCD panel because the resolutions are still not exactly matched. The TV must further scale the image to exactly 1366 x 768p so that the picture can be displayed.
An incoming 1080i/p signal must first be downconverted by the TV's internal scaler to its native 1366 x 768 resolution before the image can be displayed.
What is a video signal and how can you fit a 1920x720 signal to a 1366 x 768?
Scale the 720 input to 768, then display the 768 signal on the 768 panel.
Like if you use the LCD as your PC monitor you cannot display higher than 1366 x 768 resolution but if you (forced) output using a customized video card output settings, you will be able to display at 1080i/p.
A 1080 signal can be displayed on a 768 monitor (after scaling) only if the monitor's scaler supports 1080. On a PC, the video card scales the input to a resolution that the monitor supports.
They say that 720p and 1080i are identical in HD content/quality?
They are not identical because their resolutions are different. You probably meant that the displayed images seem to look identical to the observer.
Why is that it is better to use 720p in a PC than 1080i?
The resolutions are closer to each other. The most common resolution for computers is XGA (eXtended Graphics Array, 1024×768). 720p more closely matches 768p than 1080i. A closer match means less scaling, and less scaling usually means a cleaner, more accurate rendition.
Why is that 720p supports upto 60FPS, while 1080p is only upto 30FPS?
A manufacturer can produce a 1080p that supports 60 fps if it wants to. The specs resulted from the agreements among the manufacturers, not from the limitation of the technology.