1. What do the different coloured uniforms mean? Why are some yellow, blue and red?
They represent different departments. In Classic Trek and Enterprise, Gold represents command, red is enginneering and security, and blue is science and medical. In TNG and the other series, Red is for command and gold is for engineering and security, but blue stays the same.
2. What exactly IS warp drive? How does it work?
it's how they travel faster than light. Generally, you mix matter and antimatter, which releases a lot of energy. This energy is focussed by the dilithium crystals and sent to the warp nacellles, which create the warp field around the ship so they can go faster than light. The specifics have generally been left rather vague so they have some room to make things up for specific episode requirements.
3. Can a transporter work between different planets? Or other long distances?
Generally, the transporter doesn't have the range to beam from planet to planet. In TNG, it has a range of about 40,000 kilometers. There have been a few cases of transporters having much longer range, but Federation transporters generally don't.
4. What happened to Janice Rand? And why did she only come back for the movies?
I believe they got her out of the way so they could have Kirk going after a different space vixen each week.
5. Does Pon Farr affect female Vulcans?
I think that Enterprise established that they do.
6. If it does, do they have it at the same time as their partner?
Probably not. It's every seven years, like the guys.
7. Do they HAVE to conceive a child every time?
No. They just need a good shagging.
8. What is a stardate? How does it work? Why is it used instead of normal dates?
A stardate is a way to make future dates sound future-y. Generally, 1000 star units equate to a season/year. So stardate 42500 is a year after 41500. It's generally best to avoid thinking too much about them, because they generally don't stand up to close scrutiny. Stardates for Classic Trek were pretty much random with only a vague notion of getting higher as the series progressed.
9. If there’s no money in the 23rd century, how did Uhura pay for the tribble in Trouble With Tribbles?
She gave him a private fan dance.
10. What are those cartridge thingies?
What cartridge things? The ones Spock keeps putting into his console, the coloured squares? They're floppy disks. ;P
11. If Decker died in The Doomsday Machine, how the heck did he get to be in The Motion Picture?
That was his son.
12. Could you tell me about some medical-related gadgets and how they work? (This one is actually quite important for my fanfic)
They work because of the plot device, usually. We generally aren't told much about HOW they work, just that they do work.
13. Why do some planets have numbers after their names?
Probably because they haven't been given proper names yet and are just listed in some catalogue. We do it today, with the Messier catalogue of space objects, and the New General Catalogue (NGC) objects.
14. How exactly is the Federation related to Starfleet? (I feel SO bad for asking this!)
The Federation is the government. Starfleet is the peace keeping and exploratory service of this government.
15. Why does the Romulan race have the same features as Vulcans? They’re not from the planet, are they?
They share a common ancestor. it's generally considered that the Romulans used to be Vulcans, but they left a long time ago, rejecting Surak's teachings of logic and reason and no emotions. So some left to make their own civilisation and they became the Romulans.
16. How come Sulu’s daughter was never mentioned until Generations?
No idea. She just wasn't needed up until then.
17. How come Kirk wears a green shirt sometimes? IS NOT GOOD LOOK ON MAH HONEY.
Captains get special outfits so they can feel special and important.
18. Is Kirk the only one who calls McCoy ‘Bones’?
I think there was an episode where Spock calls him Bones, but I'm not sure.
19. If Saavik is a Vulcan, then why did she cry during Spock’s funeral? Is it really THAT sad? lol
She is half Romulan. Nothing in Canon to establish this, but when the script was written she was intended to be half and half.
20. What IS the deal with the red shirt joke? I don’t get that.
Random crewmen with names like Jones or Brown would often go down to the planet with Kirk, Spock and McCoy. And when they needed to kill someone to show how dangerous the planet was, guess who got the short straw and snuffed it?