I don't see how dogfights in space necessarily don't make sense. I think you can perfectly justify why it does make sense, no matter what anyone might object, e.g.:

Someone could just engage the FTL drive when you shoot at them and flee to where you will never find them?

• Solution 1: FTL travel requires exact calculations, hasty entry gives a 30-40% chace of colliding with a planet.
  
• Solution 2: FTL engines take an enormeous amount of energy to initiate some kind of "magic field that wraps space" or some "magic gate to hyperspace". No existing reactor can ad hoc produce such an amount of energy. Therefore, a ship entering FTL speed must charge its jump accumulator for 15-20 seconds, during which it cannot fire or have shields up. Once the "FTL magic" is established, keeping the field up takes considerably less energy.
  
• Solution 3: FTL travel creates a kind of "wormhole corridor" between two points in space through which the ship travels. Creating larger corridors takes exponentially more energy, so while distance is not a problem, the corridor cannot be made not much wider than the ship (a ship crossing the boundary to real space will be torn to pieces). This makes evasive manueuvers impossible during FTL travel. It takes several seconds before the entrance to a corridor collapses. During this time, anyone can fire into the corridor.
  
• Solution 4: Interspace zeta-distortions polarize your FTL engine, causing serious damages, frequently with class-4 radiation leaks that kill everybody on board. The Qu'ook, a race of interplanetary pirates, developed devices to generate interspace zeta-distortions, both to prevent victims from fleeing and to save them the trouble of killing the crew. Therefore, all FTL engines have a security feature which locks them down when any other ship is nearer than 2000 kilometers.
  
• Solution 5: Einstein was wrong. After the first ships that were able to travel faster than light had been built, it was discovered that light/radio waves sent from a ship travelling faster than light also travel faster than light, plus the speed of light. It is therefore possible to pursuit a fleeing ship if the general direction is known, it is possible to trace them on the radar (while travelling faster than light), and it is possible to shoot at them.
  
• Solution 6: Einstein was right, but the Nuark-kra do not perceive time and space in the same way as we do. Nor do their weapons and targetting systems which are available on the black market. Nobody knows how these weapons work, it is not even sure what they are made from.
  

Someone could travel at FTL speed and fire a dozen shots at unaware ships before they even receive a radar signal or see the attacker?

• Solution 1: Firing a weapon during FTL travel will cause the shot to go backwards through the cannon. It is not fully understood why this happens, the most accepted theory to date is that the blaster particles fall to sub-lightspeed shortly after leaving the muzzle.
  
• Solution 2: Present day targetting devices are not accurate enough for such a thing.
  
• Solution 3: During FTL travel, sensor range is limited to 20-30 meters around the ship, as the ship travels faster than the light in real space.
  

The centrifugal forces in those ridiculous manueuvers you typically see in sci-fi games and movies would kill the pilot?

• Solution 1: They actually do.
  
• Solution 2: They actually would, but spaceships are equipped with inertia damping fiels which prevent that.
  

One thing you should decide is what the purpose of the combat is which will then impact your designs for tech, combat, and other aspects of the universe.

I think that sums up pretty accurately the issue I’m having.

A future sport could work very well and lets you add a whole sponsorship, reputation, and popularity aspect to the game. After all people watch UFC, Robot wars, and heck even star craft for some reason. So why not in the future have ship to ship combat.

Ooooh, I really like the idea of sponsors. Actually, that might fill the void of a game mechanic I've been wondering for a while. See, players can enter into combat for either 1-6 hours and will automatically fight players for that length of time. Simply throwing the ship into combat and just hoping for a few lucky wins was something I wanted to discourage. I want people to keep checking in on the game through the day and fiddling with their ships to get the best possible win ratio, but couldn't think of a good way to 'punish' the player (in a very light sense) for not caring about their win/loss ratio. I didn't want to take away money or xp for loses, and I also wasn't sold on the idea of having 'battle points' like Street Fighter. What would be awesome is if the more you win the more sponsor slots you can fill up (based on level), but if you keep losing you start to lose sponsors. Having sponsors might give you special gear and give you a regular income. Maybe there is a bit of a meta game where you can't combine certain sponsors or some of them have big benifits but place restrictions on what you can do. I might be getting ahead of myself

Making a sport also gives you freedom add tournaments, rules, classes, and restrictions. You could for instance have a light circuit in which under 50 ton ship compete in or an Endurance Challenge where 16 ships compete with no repairs or reloads between matches.

Actually those are all things that are in the game already! Class restrictions, rules for certain event types etc. I've not added them to the site yet because I want to focus the testing on one area, but for example I have one zone that requires ships to be under a certain size as the combat takes place in tunnels and another that takes place near a radiation field that stops shields from regenerating.

It's funny because the more I think about it, I don't know how I would rationlize having any of these game elements without turning it into some sort of sport.

You could also let the player equip flare as part of their load out which are items that boost popularity or relationship with your sponsor but don’t give and real combat advantage.

That's a good idea!

This posts combined with Stormynature, I'm really getting sold on the sport angle...

Good points Samoth and Lorenzo, the essential feature that I forgot is that it isn't convenient to go FTL whenever you like. Reasons can be whatever; lack of accuracy, calculations needed, time to power up engines, a fixed gate needed, etc. At conventional speeds calculating a good trajectory may be the biggest factor, e.g. you know that people travelling from X to Y will pop out around a particular spot, and therefore will almost always take the most efficient trajectory passing along Z.