Congratulations, you have completed step one of problem, Identify the Problem (and you did so in detial).
Step Two: Decide whether it is a Problem that needs solving.
Being a poor DM isn't the end of everything. Some people don't have what it takes, some great DMs are really bad players. Take a few minutes and think about your situation: Do you prefer playing? Is DMing something you really want to do? Do you feel a calling to it or is it a case of having to fill the role?
Now I'll try to address your individual questions:
1. I'm constantly having problems with the investigation part of my games. Having the whole city to go looking for information seems to be to big for my players and they constantly lack ideas for finding clues.
The wizard in the party sometimes figures out ways to use his thaumaturgy to find clues but the ninja-detective-werecat character has problems using her skills to her advantage.
Another problem for me is that i'm bad at helping them figuring it out. How should i give them clues without going "yeah you just need to go to this place over there and you will find everything" - that would be boring.
I have no idea how they will try to find info so it's hard for me to plan that or to even know how they could get that info.
I have run good "hey, that's the problem" sessions and good action ones, it's that legwork and detective thing that troubles me.
Just have them roll the dice! If it's a "clue quest" then have an Investigator roll investigation and hand them a clue, if a Contact could give them a lead, have a high Contacts character make a roll!
Give them a few clues, the better the skill check the "better" the clue, and juct move it along.
I'll be honest, non-action is sometimes the hardest part to get and keep moving. Don't get discourage, just take olde timey advice: When in doubt Roll The Dice.
2. I'm lazy. I'm constantly underprepared and have to run a lot of stuff on improv. That's okay for me but the game is lacking because of that. Any tips for GMs that don't like to plan that much ?
I got nothing man. I'm a "fly by the seat of my pants" type DM and thus any prep work aside from some sketchy and often misleading notes is anathema to me.
You say "That's okay for me but the game is lacking"... is it lacking? How? If you can handle winging it, where do you feel it's not holding up?
3. My PCs right now have a job from the Summer Barons of the city to find out why some Summer Elfs attacked a group of anti-mining demonstrants.
At the same time they had to find a group of missing persons. Clues lad them to a museum where they found a sorcerer, enthraling people. After chasing him through the museum and the streats they battled a troll he had called for help. The next day the wizard conjured a water spirit to find him. The spirit warned them that this guy is the Black Miller (a fairy tale figure from our region), a sorcerer in pact with the winter court of the sidhe. (Our city is in summers territory, they try to get in)
The next session will be about them storming his home and finding proof that he tricked the elfs.
He has his home in a puppet theatre (because it's scary)
I want to try something a bit different, the session will start with them, captured and bound to chairs while the miller laughs at them. Then cut to the morning before. Should i do that or is this to railroady because they could try any kind of security measure possible and will still be captured...
That's a tough one. My advice, don't ask us... it's your group.
Some groups have no problem with an In Media Res captures scene that jumps back to "How You Got There" storytelling. Some will buck that plotline like a wild stallion. Do you know your group well enough to determine which you've got?
If yes, and you think they'll go for it, do it. If not, it might be best try that sort of move after you've learned how comfortable they are.
Also, you could just ask them if they mind starting a scene captured, knowing they have a chance to escape. If they're okay with that, they'll probably be okay with knowing capture is envitable. How you handle the capture then becomes a measure of how much being captured is the PCs idea, and how much was they were overpowered.
Anyway, i don't want them to capture or kill him because i want to use this guy in following sessions and he, as a caster, can be quite deadly to them.
I got nothing here, I usually handle those situations as they arise.
4. problem : We only have time to play 2-3 hours every week and i have problems using that time effectively. Sometimes we spend this time and not much happens in the story because they venture of in a "wrong" direction or just RP without much story happening. What can i do to make the most out of the limited gametime ?
Are the players having fun? Are they complaining about not "getting anywhere"? If no, then unless you are on a timetable (your moving soon, it's a college group, etc) don;t worry about it. The DMs job is present a story and help the players have fun.
If your not having fun then you can try to gently guide the PCs, start using a few skill checks for the PCs to discover where they've gone astray with info, etc. Try to avoid the heavy handed "NPC points them in the direction" method, use more clues, hints, etc. Avoid red herrings completely, players will naturally come up with themselves.