Ok, as for resolution, a DX lens on an FX body will ONLY give you (roughly) 2/3rds of the FX resolution. I also compare the size of the pixels on an FX and DX as follows: Think of the DX pixels as the size of a baseball and the FX pixels as the size of a softball. On a DX camera the sensor is roughly 2/3rds the size of a sensor on the FX camera so a 24mp DX sensor pixel is roughly 2/3rds the size of a 24mp FX camera. Now, as the megapixel capacity goes up the size of the pixels to reach that density has to go down in order to make room for the extra pixels. So the sensors become more expensive to manufacture. Remember that the original digital cameras were ALL DX as sensors were so expensive to manufacture and the larger the sensor the more brittle/breakable it became,that the major camera companies at the time (Canon and Nikon) elected not to manufacture FX sized sensors. They were cost (and manufacturing) prohibitive. Meaning that the sensors were so brittle that FX sized sensors would break or crack at the rate of 1 out of 3 during manufacturing and installation in a camera. Adding the cost of each sensor (FX or DX and were talking 10 mp or smaller sensors at that time) and the brittleness of sensors larger than DX, the companies decided that the FX sensors were not a viable cost of manufacture option. They would be too expensive to recover the investment. Now, as time wore on, processes and materials in sensor development allowed sensors to be both larger and more durable. They also allowed sensors to be more heavily packed with pixels. (Note, early in the FX/DX sensor development it was normal for a DX to have roughly 1/3rd more pixels than its FX brother. An FX camera might have 18mp vs a 24mp DX version of the (essentially) same camera. However, because the FX camera had larger pixels, the camera's photographic ability tended to be better for some purposes. The camera was better in low light because the pixels were larger and more receptive to light striking them. Think a 40watt lightbulb (DX) vs a 60watt lightbulb (FX). Both fit the same socket and use the same voltage but have different power consumptions. Now, many of these restrictions to FX (and larger) sensors have been resolved or worked around. Note: some of the larger format digital cameras, Hasselblad for instance, use multiple sensors placed together side to side and one above the other (like 4 panes in a window) to achieve the extra size. This is NOT because it is easier to build larger sensors but a solution to the price of larger sensors (4 smaller sensors are less expensive, less brittle, and more cost effective than a single large sensor. I guess you can thank Nikon and Canon for not putting 20 camera phone sensors into their DX cameras
Now, Canon, Nikon and other manufacturers have (as of the D610 in Nikon) been able to get 24mp resolution on an FX camera which matches the 24mp resolution of the D7100 DX camera. I think this was the first time Nikon was able to match the resolution of an FX to the (normally higher) resolution of a DX camera. Generally, the FX resolution increase was 12 to 24 months and sometimes longer, later than the release on a DX. Technology is improving on DX camera's younger brother, the FX camera. Another reason that many pros have both DX and FX cameras.

The DX lens on an FX camera is only using (roughly) the 2/3rds of the center part of the sensor, leaving the outer 1/3rd unexposed. Many FX cameras have a couple of options if you put on a DX lens.
1. they will automatically (or not automatically) sense the lens and change or ask you to change them from FX to DX mode.
OR:
2. They will expose the image leaving roughly the outer 1/3rd unexposed (note the exposed area would be round reflecting the shape of the lens, the light passing though the lens onto the sensor and not be a rectangle in the center).

An FX lens on a DX camera only uses the (roughly) center 2/3rds of the lens. The rest of the FX sized image (actually a circle image that the FX sensor would cut off to fit the FX sized sensor ignoring the parts of the circle that were outside the sensor rectangle) is thrown away.

The DX lens on a DX camera sends a circular image to the DX sensor which ignores the parts of the circle that are beyond the DX sensor (exactly as an FX lens does on an FX sensor).