In the very excellent book Spock’s World, it’s stated that Spock’s mom had to basically have IVF to get pregnant.
No amount of IVF will allow inter-species mixing. You could put a ton of eggs and a swimming pool of sperm in the most perfect womb in existence but if they’re from different species you won’t get a single conception, and trying to achieve it in a petri dish won’t change that. You can’t breed daffodils and cats, you can’t breed Klingons and Cardassians. They have completely different DNA, it would be like trying to make a boat using the instructions for a microwave.
QUARANTINE!
People, too - Wrath of Khan always bothered me because those two guys should have been quarantined. Why on earth were they allowed to just rejoin the regular crew.
plot device
Chekov even says “They put creatures into our bodies!” Why didn’t McCoy use his handy-dandy tricorder to explore this rather unusual claim?
Risk is our business mister.
Manage your risk or be managed by it.
IIRC, Spock’s parent’s reproductive material went through a lot of DNA re-sequencing (with the help of Vulcan and Earth scientists, no doubt) specifically to allow the birth of Spock. The scene in {shudder} Star Trek V was a little too simplified. Even under these circumstances, is it still impossible?
Oh, bah!
We have plant-animal hybrids today! You don’t think that in 300 years we’ll be able to hybridize Humans and Vulcans?
Besides - we are all related - there was a Next Gen episode that explained how all the major players came from a single origin.
Maybe not right now, but we are talking a few hundred years in the future. 
Although, if genetics is right, shouldn’t Spock be sterile? Mules and ligers are sterile.
For some reason, it turns out that barriers to interspecies mating don’t apply at warp speeds.
Nobody anticipated this effect. But then, nobody predicted that traveling faster than warp 10 would turn you into a newt either.
… They got better.
Didn’t Enterprise try to tackle this and end up with “it can’t be done naturally but artificially …”
I think that was the last show I watched of Voyager. What season was that?
Finally a real complaint! How many times did they call someone into the room just to show them something on a computer monitor? They call up images on the main viewer of the other side of a planet but not take a look at the same screen Geordi is looking at.
The rest of this stuff is easily explained however.
“That doesn’t make scientific sense!”
You think you know more than a academy graduate four hundred years in the future? Ha. Get over yourself.
Can’t cross breed humanoids?
In the TNG episode Emmisary Deana says “I didn’t think Human and Klingon DNAs were compatible”. The klingon/human hybrid replies “They are with a little help”. Clearly science allows, what we would call, miracles.
No security Cameras
Duh. Internal Sensors.
No diplomat ever has an uneventful voyage
Rubbish. They ferry loads of people without incident. It would just make for a crap episode so we don’t get to see it. Lets take TNG. We get less than 150 hours of episodes in a seven year period. That’s .25% of those seven years we were actually witness to.
Plenty of adventures to reuse that nifty trick they developed or let Worf earn his reputation as a great warrior. It’s even possible in that 99.75% of the time we didn’t see that Troi said something useful or interesting.
Sorry, my Disbelief Suspenders will not stretch that far.
-FrL-
This ain’t 24, though - those 150 (well, 178) “hours” weren’t in real time. Most episodes played out over several days of time instead of one hour (some were shorter - except for the epilogue conversation, Inner Light takes place over a roughly 30-minute period).
Of course, even assuming the average episode depicts a week in the life of the Enterprise-D, that leaves about half of the seven TNG years undepicted.
Yeah and did you ever notice that where ever Angela Landsbury went, somebody got murdered?
Well, speaking of defence of Bryan, Paramount has tried to rerig their “Star Date” system so that the dates can be recalculated into our current calendar system.
When you move into a new seaon of ST:TNG, the Star Dates sorta reflect that.
Edit: In other words, Paramount themselves complicated the issue by providing the means with which a fan may become confused about the length of missions, etc.
Man, I hope that’s a coincidence. That cartoonist seems way too well-adjusted to be finding inspiration on the SDMB.