First off, and I don't mean to sound preach-y or anything, but a fairly basic amount of music theory is virtually essential to your success composing and songwriting. Now, there are some musicians who can't read music whatsoever and are getting along just fine, but this works best when they are the ones playing their own music. Being able to write your music out in some form that others can decipher in order to learn it is the key to having musicians play what you want them to. Unless you are there to teach the song by ear, or if you are performing it yourself, some kind of notation is necessary, and the western music-stave approach is the most common.
So, I encourage you to learn to get the notation down. This could be as a piano part, or just a melody and chord symbols, but it will help your progress immensely. Try to learn by playing piano music or looking up pop songs you know for the tabs, see how they are written out and what makes them tick, and you will understand how to express your own music better.
Now, as to your questions about how to compose and what to start with. That's a bit like asking if the chicken or the egg came first, you get an endless loop. Music and lyrics often gets asked this, and there is no industry standard as to which comes first. Do whatever is natural to you. In reality, melody, arrangement, and structure happen simultaneously when working on a piece, to an extent. You may be working on just the melody, but you also wind up thinking of how you arrange it, and other cases like that. There are some people who are dead set in starting with one facet of the work, but most people tend to mix because it feels right for them. Chances are, some part of the overall piece will be rewritten as you realize you would like it better a different way, and you will reorganize the form or rewrite the parts because you now realize that. Think about the big picture, work on the small details, and your destination will be the same no matter where you begin.
You seem to be focused in songwriting, and I'm not entirely up to scratch in that, but I hope this helps. Don't feel pressured to write in a linear fashion. The best introductions I ever write are when I start writing the middle of the piece and come back for the intro once I've finished. Songwriting is like dreams, "you always wake up in the middle of whatever is happening". Inspiration will take you to the most important part of the song, so figure that out while you still have it. Then, once you have your parts, mess with it in chunks for structure. Once structure is put in a good place, finish any auxiliary parts you need. As you move through the piece, pull out parts of the main section you were inspired to write, seeding them throughout the whole piece. A melody here, a snippet there, and you'll look like the smartest guy in the world for pulling them all together, when all you really did was break one section apart.
For starting a piece, or finding a melody, my best recommendation is to improvise. If you can learn pieces by ear, you can improvise just fine. As you come up with lots of bad improvised melodies, just remember and write down the good ones. Repeat chords as you try this, until the background of the song is hardwired in your brain, and the piece should begin to work itself out.
On a final note, I'd like to welcome you to the forum. Also, though you don't seem like a classical person as much, I'd like to recommend you listen to some minimalist classical music. You mentioned you like electronic dance music? Minimalism like Reich and Glass is actually very similar, and maybe something you'd be interested in. These are the kinds of people who can write a 30 minute piece based on one of your 8-bar phrases and pull it off. Check it out a bit, see if it works for you.