I would completely ignore the effects of faster-than-light travel because it is not fun (to most people, anyway) and doesn't add any real value to the game (If you want time travel, you can add time travel devices, and it doesn't even matter how they work! Anti-buzzon inverters if you want, or gremlins, or simply "unknown alien tech".) but being too realistic adds considerable trouble and spoils fun. I would also make communication (radio, video conference or "holonet" or whatever it is) faster than faster-than-light. Who gives a crap, it has to work or people are unhappy.
Also, firing your lasers on a FTL ship should cause the laser beams to travel faster than light, and faster than your ship. But generally, they should be slow enough so a human can see them (how do you see them anyway?) and do an evasion manueuver.
In Star Trek, they don't waste much of a thought on the speed of light (though in Voyager, they do, somewhat half-assed, and in one episode of DS9 as well, and of course there is the famous "Picard Manueuver"), nor in Star Wars. You can call someone at the other side of the galaxy no problem, and have a nice conversation without any lag. After ending the conversation, you can even see the other starship which is in a different solar system from the exterior when there is no camera to record this picture at all! Great stuff. It's not realistic, but it also doesn't suck.
Imagine Picard is calling Starfleet to tell them the Borg are coming. Too bad it takes 8 weeks before the other end to turn on the monitor and says "Hello? What's up?", and another 8 weeks before the "Hello" arrives. And he still hasn't told them a word. This sucks. How do you fit this into a 40 minute episode?
It is "somewhat OK" if it adds something to the story, such as in this DS9 episode where that woman captain which Sisko gets involved with wants to dearly see the recording of this baseball game her brother played, and the signal transmission takes 3-4 weeks since it's as far away from Earth as DS9, but in the opposite direction, blah blah. It is a nice plot device because it gives the couple an occasion to get together on something special that they both like. Which is, of course, what matters for the story. Except Sisko can talk to the guys on Earth (including his father at home, so it is not some super secret military special technology) in realtime without any delay at all, so... bleh. So much for consistency.
Also, story and "game fun" apart, what we "know", or what physicists say isn't necessarily true. If you're being realistic, they're just telling you what they think, not what is truly the case (though incidentially those might be the same thing). It is true that so far no evidence has been found that would dismiss the idea of time dilation, but there is no verifiable hard evidence for it either. Yes, one or the other petty thing has been measured here and there, and it doesn't contradict, but truth is we know absolutely nothing about travelling anywhere close to relativistic speeds, let alone faster than light. Even if all assumptions for "close to" are known to be demonstrably true one day, this still doeesn't extrapolate to "faster than". Until we do travel faster than light, it's all just talk, and talk is cheap.
Many of the models and arguments (such as e.g. about causality) aren't necessarily binding. For example, who told you that there is any such thing as causality in the universe or that it has to be "fixed"? Maybe the universe is perfectly happy with things that you deem paradox or impossible.
Measurements are not necessarily binding either. Whatever you think you measure might be an entirely different effect. And, what's true for a sub-atomic particle (or for light) might not at all be true for your spacecraft and your body within.
Let's say you an atomic clock in a shuttle and have it go around the planet a few times and compare it to another one. Wow, they're different. What does it tell you? It tells you that you cannot dismiss the time dilation theory, but other than that it tells you exactly nothing. You have no idea whether the atomic clock did fewer "blips" because time was dilated, or maybe because the acceleration then the shuttle lifts off causes the atoms to "bang together" in a particular way. Or maybe cosmic radiation has an influence, or something else. Gremlins!
But even assuming that what you've measured is absolutely conclusive without any possible doubt, it only tells you about an effect that happens at pathetically slow speeds. The fastest (unmanned) spacecraft in history has reached 0.02% of the speed of light. You cannot possibly know if the same effects that happen at 0.02% are still happening at 50% or 90% or 99% (or at 150%!). Maybe matter transforms into "ether" at 99.99% the speed of light. How could you know for sure that this isn't the case?
