I was quite superficial in my explanation, I guess. There are many different ways of mediation. The word really is just a term for “mental exercise” as opposed to physical exercise, if you ask me. And that is a very broad meaning. What I am specifically doing is actually a relaxation technique I was taught once. I focus on my breathing and count up whenever I breath out, up to 4. Once I am at 4, the next breath restarts at 1. It is really simple but actually very hard to keep up. Why? Well, because all those thoughts keep invading my head and take my off the trail 
That’s why it is an exercise and not an end all by itself. There is so much more way for improvement. The goal is to acknowledge all unrelated thoughts and let them drift off again without occupying too much of your attention (but not specifically blocking yourself from random thoughts!). This has helped me fall asleep very successfully many times. I also apply this technique when I wake up in the middle of the night and have trouble falling asleep although it is much less effective then. Sadly I don’t know why.
I learned of this one in a seminar about relaxation techniques. We also tried Qi Dong, Tai Chi, progressive muscle relaxation and many others, but this breath counting one turned out to be the most effective for me. But in the end almost every participant chose different techniques that worked best for them, so I suggest you also try to get your hands on many different ones and find one that suits you well. Since you describe your problem as being a mental one rather than a physical one (same for me), i suggest you try the mediation ones first over, say, techniques that involve physical exercise.
A note about meditation making your more wakeful: The same breathing technique that I use can indeed also be used to become more in tune with your surroundings in the waking world. Many years after I learned it as a relaxation technique, I also stumbled across it again in the context of Mindfulness. It depends on your inner focus while practicing the technique. I find this very fascinating, because it feels there is so much still to learn.
A classic! Don’t you worry. Most people encountered this problem at the start and some still do many years later 
I found the most reliable way of staying calm is to condition your mind. Again, this takes time and practice. You can get the best results by working on your attitude constantly, a little bit every day will do wonders. What really should help is to focus on either your goal or your plan of action. There are different types of people. Maybe have a look at this thread to find out if direct (goal) or indirect (plan of action) would work better for you.
Here’s something I do every night. After I go to bed, I say aloud to myself what I will do once I become lucid. It has become like a mantra that I could evoke even asleep. And that’s kind of the point, you see 
So when I become lucid, I know exactly what to do. This greatly helps to reduce excitement and stay calm. But the same can also work with the techniques. Go through the tutorials step by step and make sure you absolutely understand what is supposed to happen to your body and mind and the preparedness will usher in more calmness while you are doing it. That’s the best advice I can think of.
Now you may wonder, “but will being completely calm make it less exciting of an experience in the end”? Well, to the ignorant outsider it would definitely seem that way, but I can tell you that even the most dull lucid dream is still an amazing otherworldly experience unobtainable by other means 
There are many intriguing things you can learn here - and not just about your dreams.