I'm happy to take a stab at your questions. Maybe reading what Abra said and then what I say will give a more complete answer.

1) Dream Recall:Many people have very choppy dream recall. It even happens to me from time to time when I'm being lazy about it. A few things help me go from a scene or image to most of the dream. I lay in bed in the position I woke up in and just let my mind drift. After a few minutes more of the dream, or even a dream I'd forgotten completely, will come back. Another thing that works for me is to look at what I do remember and then to just let my mind drift with that as the starting point. Ususally that will give me the whole dream in reverse order from that point. Sometimes I'll even remember a dream or two from before it. I've also taught myself to become aware of my night time awakenings. This way I just jot down quick dream notes several times during the night.
Dream Signs:Being in my old schools is a common dream sign for me too. You can practice recognizing this dream sign in this way: Imagine you are at your old school (or better yet remember a dream where you were at your old school). Try to make this imagining as vivid as possible. Then think to your self some thing like, "I'm not in school anymore. I wonder if I'm dreaming." Try to imagine how it will feel to think that in your dream. Then do a reality check. A tutorial on RCs is here. While doing the RC imagine what it will feel like to have a RC tell you that you are dreaming. Imagine what you will do when you realize it really is a dream.

2) Almost everyone who is experienced at LDing will agree that the "best" LDs are more vivid than real life somehow. Hyper-real. I've had some LDs where I cried because everything was so overwelmingly clear. I sometimes have LDs where I think about what I was doing last night, what I'll do in the morning, etc. However, I mostly focus on the dream. I'm not sure if it is because I don't recall my waking life very well or if it is just because I like to get caught up in it all.

3)
a. The sucked into your bed sensation you described is a fairly common sensation associated with hypnagogia. Other common sensations are falling, rocking, vibrating, shaking, tingling, hot and cold, pressure, and rapid heart beat.
b. The music, if you are hearing it "outside of your head"(as if your ears are hearing it) instead of "inside" (as if you were imagining it) is also hypnagogic hallucinations (HI).
c. These quickie dreams can be one of two things. They may be HI, or they may be what is known as "non-REM dreams." Most people don't remember their non-REM dreams. They are very similar to HI, except you are actually full asleep and they tend to be more multi-sensory than HI. Many of my non-REM dreams are about a setting, a person, or an idea but they lack a plot. Generally I'm simply observing, and don't seem to have much of a body to move. Because of the lack of plot, there is no need to move. This is a good thing, since the physical body is not under sleep paralysis during these dreams.
d. The twitches you describe are probably "hypnic jerks." These occur as your body slips into or out of the sleep state. They can also happen as your body comes out of sleep paralysis. Hypnic jerks are most common when you are overly tired or during the night.

4) As Abra told you, FAs only happen when you are really asleep and dreaming about waking up. It is probable that some of the instances you describe were FAs and some where just confused, slow real awakenings. I tend to get a lot of FA if I'm sleeping in in the morning. I will wake up for real briefly in between some of these FAs. Some of them turn into dreams about getting up for the day, some become LDs when I realize I'm still asleep, and some come and go rapidly, as if they were non-REM FA dreams.

5) It sounds like you are very promising. I'll bet once you get past your conviction that everything is real, you will be very good at LDing! I suggest, if you want to learn to LD, that you start to practice. By that I mean that you should think about past dreams you remember well. Your learning to fly dreams would be perfect for this. Remember them as vividly as you can. Try to relive the dream. Then imagine that you think at some point, "I wonder if this is a dream?" Imagine vividly how it will feel to say this to yourself. Then do a RC. Then imagine who it will feel in the dream to realize you are dreaming. Think about what you might do now that you it is a dream. Do this over and over, with as many remembered dreams as possible. This exercize will help your mind to learn to doubt reality, and to know what to do when you aren't sure.

I hope my perspective helps!