1. As Trond said the writing is an individual thing, but basically the difference between a melody and a improvisation is that improvisation is something common, something that will not draw your attention much because you heard something similar many times and it's just a bunch of phrases, while melody stands out, has some kind of meaning and it's original. So basically you need to find out what do you want to write, what do you want to say, what emotions to express and let the melody come out from your head, not from your fingers
2. The best is to learn notes, tabs are not so good since you cannot write the length of each note, just the pitch. Or to get some software like Reaper and to record all the ideas.
I don't know how you function, but I cannot tell to myself that I will write a song and to do it. The melodies usually come itself. Some weeks ago, one evening, I was very tired and got to bed, and one melody started to play in my head, so I turned on the light, took a pen and paper and wrote it down because I knew in the morning it would be lost if I don't do so.
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