Yes and no. A 20 mp FF file and a 20mp APS-C file should be roughly the same size and contain the same amount of information. Because the APS-C is sampling a smaller image size the lens won't usually be able to resolve as many lines as the FX. If you look at DxO lens testing, you will notice that FX lenses can resolve 25 to 30 MP where as APS-C lenses rarely resolve more than 14mp and in most cases it is like 6-10. And, if you compare the resolution of the same FX lens on APS-C, it will usually show much lower lower resolution numbers. There are darn few lenses on APS-C that can take advantage of the 21-24MP sensors on even entry level dSLR's
Nope.
DxO is not worth discussing. Bunch of ignorant technocrats who only look at their equipment and measurement results but never look at actual images. For example bitingly sharp lenses get called soft, soft lenses get called sharp, all because they are unable to even test something trivial as lens sharpness without doing every n00b error in the book. And thats sharpness, which is relatively easy to test.
And yes DX lenses are usually pretty poor. Thats because of how they are made - cheaply - not because theres something magic about full frame lenses thats suddenly gone with APS-C. Besides one can always use full frame lenses on APS-C sensors, anyway.
And no an APS-C sensor will not record the same information as a full frame sensor. Otherwise why would we bother buying bigger sensors ?
If you offer me a D4s with 16 megapixels and a D7200 with 24 megapixels as a constant free loan (so I cant sell either) I'll pick the D4s. Unless I need the smaller pixels.