GRs interactions with Asimov are well documented, and the examples here are mostly to do with character and storytelling instead of science.
Whoever wrote #7 on that list is seriously confused between light years and calendar years and probably doesn't understand general relativity either. Bad science... pay attention to #7:
http://hotnerdgirl.com/tag/16-facts-you-never-knew-about-gene-roddenberry/
The DeForrest institute's involvement with TOS is also well documented, but there are some fun examples here of what they picked up (and what was ignored).
That quote is Roddenberry's attempt to explain away SDs incobinconscies with a "fudge". He admitted elsewhere he doesn't even know what it means. It's certainly not a magic formula used by the writers[ "This time system adjusts for shifts in relative time which occur due to the vessel's speed and space warp capability. It has little relationship to Earth's time as we know it. One hour aboard the U.S.S.Enterprise at different times may equal as little as three Earth hours. The stardates specified in the log entry must be computed against the speed of the vessel, the space warp, and its position within our galaxy, in order to give a meaningful reading."] - Roddenberry
Stardates are not 'Dates, per se rather a relative position in the galaxy at the time of Log.