I'm an avid reader of /r/DaystromInstitute where all kinds of things about the Trek universe are discussed.
Yeah, there are some outlandish things, but things that you know go against physics they at least usually have some technology that they mention that circumvents it. They acknowledge the issue. "You can't travel faster than light!" Right, and the ship doesn't actually ever exceed c, it's in a bubble of warped space. There were writer manuals on what you can and can't do within the Trek universe. The transporter even has a "Heisenberg Compensator"
Of course I never really watched much of TOS, I was more TNG era and up.
But yes, B5 was also an enjoyable show, I wish they still had it on Netflix.
Generally speaking, when they invent a tech you're absolutely right that they come up with some kind of McGuffin to keep it consistent. The ones that I found the funniest though where things like the crack in the event horizon - treating a line of demarcation like it's a physical shell was just ... weird. The left handed neutrino one was a DS9 episode where they realized Quark was manipulating probability in some space roulette game (can't remember the name) and the proof was that all the particular particle had a left handed spin. The problem with it was that while it's true particles can have a left and a right handed spin, the particular particle they settled on was one which always, 100% of the time in nature, has a left handed spin.
The warp idea - yeah, I get the idea of creating a bubble of normalized space that essentially can avoid violating c because it's not acceleration - it's more like a lubricated thing moving through whatever. That's fine. But a stable wormhole connecting two specific points in space/time (Stargate series) or non-adjacent hyperspace (ie moving 15 feet to the left in that space, then reentering normal space does not equate to 15 feet in our space, but to a completely different location) are the more likely options from a science POV. Einstein's math supports a non-adjacent hyperspace as being theoretically likely. Wormholes already exist naturally - they just don't exist for more than a blink of an eye and they don't get much larger than a pencil sized hole. They spontaneously pop in and out of existence, spew a few particles and go away.
Don't get me wrong - I love Star Trek. Haven't watched Enterprise, haven't seen the last 3 seasons of Voyager (yet - been burning through it marathon style with the wife), haven't seen more than a handful of DS9 episodes after season 2 ended, but I've seen TOS, TNG, the movies, etc. countless times. Am an avid Star Fleet Battles lover and know way, way, way too many facts about the show's lore and all that. Absolutely great entertainment and as DaystromInstitute proves, it's something that will spawn advancement in technologies simply because it gets people thinking about 'well, how could we do that?' Cell phones are something we take for granted, but if you think about it, you can trace the idea for them back to the early 20th century when someone wrote a comic called Dick Tracy. His wrist communicator was the first place I've seen that presented the idea of a portable phone that didn't need wires. SciFi is awesome for that and spurring the minds of folks to create wild new ideas.
***edit: holy cow that's long. even I say TLDR, skip this one