This is not correct. Dragon was in production before the NASA contract was signed. SpaceX was paid to develop a commercial crew variant called Dragon V2.
Spacex was from its start conceived by Musk to develop low cost systems that will be used to colonize Mars. He has said this from the start.
Nasa contracts were never necessary for SpaceX to succeed as they have contracts world wide to lift payloads to space for many nations. These contracts help to accelerate the company timeline for crewed missions to the Moon and Mars.
Sort of. According to Wikipedia the plans are for the Dragon to be modified into the "Red Dragon" for future mars missions.
"Red Dragon is a modified SpaceX Dragon capsule for low-cost Mars lander missions using Falcon Heavy rockets. Plans call for a sample return rover to be delivered to the Martian surface while also testing techniques to enter the Martian atmosphere with equipment a human crew could eventually use."
Note, for all you "SpaceX and NASA are competitors" people, that the Red Dragon design study was something that NASA paid for.[1] SpaceX and NASA are are on the same side in reality. Only in magical internet flamewarland, carried out by people who don't do any of this for a living, are they in opposition.
It would make more sense to contrast SpaceX with ULA or any of the other rocketry startups. Whatever long-term plans SpaceX have aren't very relevant to the current supplier-contractor partnership that SpaceX and NASA share.
You point is correct, but SpaceX is competing directly with the SLS which according to Wikipedia is designed by NASA. I am under the impression that the ULA will produce the SLS, but after a quick search I can only find reference to ULA producing the upper stage (http://www.ulalaunch.com/united-launch-alliance-hosts-nasa.a...). NASA as a whole is not competing with SpaceX, but some parts of it are, unless "designed by" really just means "funded by".