Well, you're micro managing is largely dictated by your fleet mix and strategy behind that fleet mix. I'll address your questions as best I can, and I'll give you some examples of what I've been doing. Now I've only been playing for about 3 weeks, so anything I say might not be optimal. I consider myself a decent player (I've played against humans online twice and won both times, but that's enough games to call myself any kind of an expert), but there are certainly better players than me.
In general, I just assign my entire fleet to a control group, but I usually don't designate them as a fleet. The whole deal with designating a fleet is that it causes all the ships in the fleet to stay in either tight, standard, or loose formation with the flagship. Sometimes I will fleet flak frigates together. I tend to not micro manage them as much. I will usually just position them somewhere that I think enemy fighters will pass as they make their attacks so that they can catch the enemies as they flyby.
In any case, I generally don't fleet my ships, just control group them. In a battle, you can hold the alt key and click a unit in the empire window and grab all the ships of that type if I want to task certain classes to firing on certain enemies.
I usually switch off the automatic special features on my cap ships and command them "manually", as well as trying to direct focus fire against important targets.
I do this for the most part, but there are some special abilities that lend themselves well to automatic use that I leave on. Two examples from the TEC would be the Marza's siege guns ability and the Kol battleship's gauss rail cannon ability. Most area effect abilities or ship disabling abilities (like TEC Akkan's Ion Bolt or Vasari Evacuator's Gravity Bomb, or Advent Mothership's Malice ability) I leave on manual. The AI is not too good at intelligently using the area effect abilities, and disabling abilities like Ion Bolt I want to keep handy so I can pop that retreating capital ship that I've almost got destroyed.
BUT I have no clue what I should do, for example, with the repair cruisers.
I usually leave my support cruisers on automatic and let the AI handle them. The TEC's robotics cruiser lends itself particularly well to automatic use, IMO. With micromanaging, you've only got so much attention span. I find that it takes everything I have to manage my cap ship abilities and designating targets for certain classes of ships in my fleet. I just don't have enough left over to manage support cruisers.
What about the strike craft?
Again, depends on fleet mix and strategy. I like to use a 50/50 combo of fighters and bombers in the early and mid game and I do micro my strikecraft some. I generally keep my bombers docked at the beginning of a battle because enemy interceptors usually go after them first, and bombers are dog meat for interceptors. I micro my own interceptors to suppress the enemy interceptors. This can require some attention because your interceptors will target enemy bombers by default, so often times you have to get hold of your interceptors and re task them. Once I have enemy interceptors mostly suppressed, I launch my bombers and I'll usually focus fire them on either enemy capitals first or heavy cruisers if he has a lot of them. I'll focus fire my light frigates against enemy carriers to prevent enemy interceptors from gaining strength again. I'll usually focus my capital ships against enemy capital ships and micro my own capital's abilities to best effect.
I also don't really know what cap ships' special abilities are really useful and which aren't worth it.
Again, it's all dependent on what kind of strategy and fleet mix suits you. I'd be happy to tell what kind of caps I usually get and how I balance their abilities, but this post is already pretty long. I could send you a PM if you're interested though.
In general what I do is get a capital that can colonize as my first capital no matter which faction I'm playing. Besides being able to colonize, these ships usually have good special abilities, especially the Advent mothership. After that I try to choose capitals that will best augment the type of fleet I'm planning on building. Again, if you want something more in depth just let me know and I'll PM you my little treatise on capital ship tactics.
I've the feeling that during the battles I'm not really commanding my forces very efficiently...
It does take some practice. It helps to be able to quickly identify where your fleet is weak and then identify which ships opposing you are going to be the biggest threat and eliminate them first. For example, if your fleet is mostly light frigates and your enemy has some LRMs, task something to take out the LRMs first since they are really hard on light frigates. Interceptors are good in the LRM suppresion role, BTW.