First, your description is accurate. The only pedantic critique I would make is that the technical term for nucleotides in DNA is deoxyribonucleotide.
Second, I don't want to say that non-helical DNA never occurs since the structure of any macromolecule is dynamic, but I am specifically avoiding exceptions to the rule to avoid confusing the issue.
The helical structure of DNA is a low energy form which makes its formation thermodynamically favourable. Chemical bonds in DNA (and every molecule) have conformational flexibility which means that the molecule as a whole can adopt different structures. If you picture two DNA strands joined by hydrogen bonds but in a non-helical, straight form, you can simply twist the strands about their central axis to form a helical structure. This twisting is allowed because of the conformational flexibility of the chemical bonds. The helical structure is more stable than the "straight" form (because of base stacking interactions), and so it forms spontaneously. I struggled in vain to find a good animation of this, but came up short. You can skim through this video to see an example of what I'm talking about. It's subtle, but you may be able to see, or conceptualize, that as the two strands are twisted around each other, the adjacent bases in a strand come closer together. This permits stabilizing interactions between the adjacent bases which favours spontaneous helix formation (I briefly discuss this at the end of another answer).
As for how frequently the "rungs occur", this is dependent on the specific helix geometry. In B-form DNA, the geometry thought to occur predominantly in vivo, the helix completes one full turn approximately every 10.5 bases and the spacing between adjacent bases is ~3.4 Å, as shown below: