Maybe this will explain some of the questions you have about Warp drive and speed;

Warp drive & speed; [http://memory-alpha.wikia.com/wiki/Warp_drive]

All of those things are full of inconsistencies, retcons, and even contradictions. But even just within that same episode, they wouldn't even get to Proxima Centauri within the time frame they claimed they could get to the Klingon home-world.

Indeed, there's good evidence in the TOS episode "Metamorphosis" - including what Spock says about him - to argue that Cochrane invented/discovered warp drive not only for earth/humans, but for EVERYONE. And this is not directly contradicted until the "First Contact" movie.

Like I said it's just a show be glad that they at least "tried" to explain the futuristic technology. And inconsistencies are bound to happen in a franchise that is over fifty years old in the case of Star Trek there are numerous but aren't glaring inconsistencies like in Star Wars between the OT and the PT.

Sure, but I have more of an issue - and I think it's understandable - with the contradictions even within a single episode, not over a few decades. It would be along the lines of someone in a movie or tv episode talking about how they live in New York, their car goes 65mph on the freeways, and hey let's drive to LA for lunch and drive back to New York in time for dinner. It would take DAYS, not hours, by the very measurements they just gave!

Someone writing for Star Trek, let alone a producer etc, should be smarter than to say they can get to Neptune and back in a few minutes, so naturally Alpha Centauri is just a few hours. And the Klingon homeworld, heck, just a couple days!

And your basic viewer is not a scientist so you have to "dumb down"things a bit for them to understand or make it more acceptable for them.

I guess. Not sure what difference it would make to say it takes a while to get to Alpha Centauri even at warp 4, because it's not like they're going to SHOW it all in "real time."

In the Voyager episode "The 37s," Tom Paris says that warp 9.9 is about 4 billion miles per second. That works out to about 21,500 times the speed of light. But the cube root of 21,500 is 27.8. So they remained inconsistent of how warp factors worked.

It's just a show just accept the inconsistencies you'll have far more time enjoying the show they just named a number which sounded really big and would sound plausible for the average viewer.

Star Trek was also intended to attract an above-average audience.

Therefore the effort to make it sound plausible and that is the keyword here plausible, and to be honest the shows after TOS were getting a bit too heavy handed in the science department where technobabble became the new "Deus ex machina". Sure intelligent people might like the show but also know that most of the science and technology depicted is fairy tale stuff (in such a case it would be called magic but in the case of Trek technobabble) and why you have such a problem with the Warp drive the transporter is a technology far more difficult to accomplish and how it works relies heavily on fiction. To give an indication if one could digitize a human body for transportation the storage capacity needed to correctly store the place of every atom in a body(not withstanding the Heisenberg uncertainty principle(hence the Heisenberg compensators)) would be larger than the Milky Way.

Yes, if the transporter were actually recording etc, then it could be a problem in that way. But, for example, a radio signal or TV signal can be transmitted and received without the need to account for every single atom involved. Which is still technobabble in a way, of course. But the bigger problem with the transporter - or two problems, actually - are whether the process actually involves moving the individual or creating a copy, and pretty much either way, the basically impossibility of ever ending up with TWO of someone which happened in both TOS and TNG.

Not only in TOS and TNG but in VOY as well (where we had the whole Tuvix thing)

Tux wasn't a duplicate the same way, though. It combined two beings into one. Although it's pretty much the same issue: if Tuvix didn't weigh as much as Tuvok and Neelix combined, where did the extra mass go? And where did the mass come back from, when they were separated again?

Stop overthinking every thing as if it was based in scientific reality coz it is not.